Modern Vespa

i have Vespa 125 2013 it keeps cranking when battery is connected and if the battery is connected and without the key also it keeps cranking and when i turn the key it does start

The starter relay is stuck. It should be easy and inexpensive to replace.

Can I do it myself? If I can how?

You can probably do it yourself. I don't know the location of the starter relay on your scooter (What model is your Vespa?) , though it's usually easily accessible on most scooters. You could probably find a parts diagram for your Vespa which would show the location, or you could search the forum for more about how to change a starter relay. Good luck!

It's a vespa 125 that's the model name written on the vehicle papers

That's not enough to go on. What year is it?

If it's a modern Vespa, it should be an LX or Sprint or have some other model name.

(A picture would help here).

It's 2012 vespa 125 and the registration picture I am attaching here and if you want the vehicles picture I can send to you tomorrow I can't take pictures right now because it's midnight here

For 2012, that's likely to be some variant of the Vespa LX 125.

Here's a thread on the same problem you are reporting, on what I think is the same (or similar) model:

lx150 starter problem

Thank you

Can I start vespa without a carburetor like by giving fuel directly to the engine?

I would think a 2012 is fuel injected not carb. How did you go from a possibly stuck starter relay to wanting to dump gas directly into the engine??

fixed the starter relay but now it's not starting it's cranking but not starting it's hasn't been start since 1 or 2 years ig

You can squirt a small amount of gas into the intake. If the motor runs for a few seconds, you've got a fuel/carb/injector problem.
BTW, have you checked all your fuses?

Yes I have checked the fuses and they all fine do you have any pictures or video so I can refer ?

As I recall, there has been a carbed version of the small frame Vespa built in India in recent years so it would be good to confirm carbed versus fuel injected.