Modern Vespa

So the 2018 Vespa GTS 300 is misbehaving. It starts intermittently, i.e. wake up and go and start it will start fine, go ride it a few miles. Park it go into a coffee shop and come out 15-30 mins later, it wont start. Leave it there for sometime (that seems to vary) and then start it, it works fine.
Yesterday for 4 or 5th time now in the last 4 months it misbehaved, had to leave it overnight (rolled it to a spot where it was safe till Friday for alternate side parking rules). Went over this morning, no problems started and everything was fine.
When this problem initially started back in June/August 20, I checked(cleaned the points, battery, the brake switches, and cleaned the starter and the kill switches, put some dielectric grease and thought that had fixed the problem. But it then started to misbehave again late December into this new year.
My thinking is that the starter relay is the culprit in that it may have some corrosion/carbon in the pins from sparking possibly and when the Vespa warms up it makes those pins expand enough to cause a loose contact and thats why it doesn't start, but when it cools down it contracts and the contact is reestablished.
If that theory isn't a viable reason for this misbehavior what could be causing this MISBEHAVIOR.
Before I order a starter relay and replace it, is there something else that is that the culprit in all this.

I take it it's not even attempting to turn over given you're looking at the starter relay.
Give the starter motor a gentle tap with a hammer next time.

Put your spare starter relay in - yes, it's one of those spares that are cheap but frustrating not to have at hand when needed! Same with a pair of brake light switches, exhaust gasket etc...

Yes it wont turn even. Just stays silent after the fuel pump whirs, I haven't heard the starter relay click when it generally starts.
But as Jim states, I am making the purchase of the relay and maybe I'll throw in those brake switches in as well keep them handy cuz it will save on shipping.
Muchas gracias.

Check for 12V on the motor too.

https://modernvespa.com/forum/post2572879#2572879

Fuel pump, or vapor lock issue.

Irrelevant I'm afraid. It doesn't even turn over, as stated above.

Any of the switches in the circuit can affect this. Kill switch (and side stand switch if fitted), starter switch, brake light switches, starter relay, fuses, rodent attack, whatever - but something is open air where some copper should be.

The OP posted the following;

It starts intermittently, i.e. wake up and go and start it will start fine, go ride it a few miles. Park it go into a coffee shop and come out 15-30 mins later, it wont start. Leave it there for sometime (that seems to vary) and then start it, it works fine.

This screams fuel pump and/or vapor lock to me. Been there before with my Piaggio X9 Evolution and Yamaha T-Max. The heat from riding is a sure sign. Once cooled, the vapor lock dissipates and the thing starts.

A fuel pump or vapour lock issue would at least allow the engine to turn over.

Areed, and I do see that the OP later revised his issue to say he now has a no crank situation which is something else. Not too sure what his issue(s) really are so cannot offer any useful advice. Cheers

To be a bit more clearer. When it doesn't start, I mean it doesn't even turn. The battery is fine and the switches are fine but as Jim said where there should be some copper there is some space and that causes the circuit to not be complete. Since the brake switches are working (brake lights light up) the fuel pump whirrs, there is no side stand so no switch there, when the key is inserted and turned on the speedo does its check and all systems are ready to go except there is nothing. I try toggling the kill switch, remove the key and re-insert etc to no avail. Leaving alone for a while seems to have done the trick thus far whenever this has happened. I have had to leave the Vespa at some place & come back the next day and it has started up fine.
I have ordered a starter relay and will replace that when it arrives. Hopefully that will solve the problem.
Will update when I replace the relay unit and we shall then see if it has resolved the issue.

Use a multimeter or test light to see if:
12V gets to the starter relay coil [1] - if yes then the other connection goes to 0V when you try to start. [2].
[1] You need 12V at the relay coil for any hope...
[2] The 'ground' connection to the coil is from the ECU. If not there, then the ECU is duff (very very unlikely) or is being given duff info (kill switch etc).

If all the above is good and there's no turning - check to see if 12V gets to the starter itself (actually not a bad first step before the above) and if so, you have a duff starter - not unknown for them to not like being 'hot'.

I am interested in seeing this post to the end.

Checking the voltage at the starter motor involves taking out the pet carrier. On the Primavera at least, the relay is under the floor.

Update to the problem I have been having.
So about 10 days ago, the Vespa misbehaved and wouldn't start after I had gone to some place and left it parked for a couple of hours.
I had to push it home about 4.5 miles in all, luckily only about a mile of it was uphill. I had some sore thigh muscles for the next two days.
I left it parked, ordered the parts I was supposed to order i.e. a new starter relay along with some regular maintenance stuff like a new belt & i decided I would change the rollers for some Dr.Pulley sliders.
I got everything and today is relatively warm, I decided I would work on the Vespa starting issue at least. Remember, it would start but then decide to misbehave and not start again for sometimes a few hours other times overnight, I had concluded with expert MODERN VESPA FORUM advice that the relay unit was most likely the culprit. Before I changed the relay unit I decided to see if it would start after the 10 day hiatus, and it did not start. So I replaced the relay and the result was the same. Doesn't start. The brake light works so it isn't the brake light switch(both front & back) work. The horn works, the kill switch has been toggled on and off.
So now I am not sure what to do. Maybe Ill try using the Master key and see if that fixes the problem if not then what else do I tackle.
I have lent my multimeter to a friend so won't get it back till tomorrow, then Ill check the voltage on the battery, though horn was strong, still will check to see what the voltage is.
What do the wise folks advise I do next to resolve the issue.

OK so the weather was warm yesterday & today. I managed to take some extra time off during lunch and checked out the Vespa situation.
Checked the voltage on the battery, it was 12.59V with no load. Then I decided to connect the battery tender to charge the battery up. Charged it up in 30 mins to around 13.9V. Tried to start and still wouldn't start. Put in the master brown key and still no starting.

Then I remembered that jimC on another post about testing the starter relay.
So I put my multimeter into the prongs on the starter relay and tried to see if there was a fault in the starter, but instead the Vespa started up.
So i removed the multimeter, turned off the key, turned it back on and tried to start it up again, it flinched but then started up fine.
So to the dumb me it seems like the connectors going to the relay are slightly loose. I used a plier to slightly crimp the connector. It starts up now but there is a slight hesitation and then I have to press the starter button again.
Maybe I need to crimp in some new connectors.
Any of the wise folks want to educate me on whether this is the cause or is there something else that is wrong and this starting was a fluke only.

Sounds like you're getting closer. There could be more than one thing wrong. Before I started replacing connectors, switches, etc. I would go back and check the battery again, under load while it's trying to turn the starter over, just to rule out a bad battery. Good luck!

Will check as per your suggestion over the weekend. Just got free from work, went and tried to start it, it went back on strike & wouldn't start, used a plier to kinda crimp tightly the connectors no effect.
Just hope it is a battery that is just dying, it has been in there since 2018 October. Maybe the use of heated grips over the last 3 years has put an added load to make it deplete faster.

Ok tried to get the Vespa to start again, battery was reading 12.45V and no go on the starting. So put the battery tender on for an hour to see if the battery would charge up, it would not get past 12.7V after several hours getting it to charge on the trickle charger.
So couldn't get to test the battery under load as it wouldn't even turn.
I will order a new battery and see if that resolves the issue.
Fuses are all ok, the starter relay is ok, the brake light switches are ok.
There is enough juice to get the horn to work along with the fuel pump to come on and the speedo does it's diagnostics.
So question is if 12.45-12.7V is enough to get it to turn at least and if it doesn't is it something else or is it the battery at fault?

A battery can easily have enough charge to turn on the lights, even sound the horn, but not enough juice to make the starter turn the engine over. If you can measure the battery voltage while the starter is turning you might see it drop several volts. Not saying for sure your battery is bad, but it's suspect.

It may be the brushes in the starter motor sticking. Give it a tap with a hammer next time it won't start and see if that gets it going.

While you are tapping the starter motor with a hammer, tap the sidestand switch.
I assume the brake light is on.

Brake light is on. Ill check the starter motor, though when I got it to start the other day it worked fine a few times but then, when I tried starting it again a few hours later it wouldn't do so.
The battery is 5 years old, I am going to have to replace it soon anyways, so I shall just order one and then go from there.

^ depeding on mileage it might be the brushes in the starter that are stuck or worn.
In theory you should hear or feel the starter relay operate.

Don't over think it and do the easiest fix first.