Modern Vespa

Tilt lock is not functioning properly; reservoir is full; horn beeps 5 times when engaged.
Previously had oil leak on ground which has since stopped.
Bike stays upright but can be leaned left and right when locked.
Dealer suggested issue(s): The leak will be either the tilt lock sensor or the hydraulic master cylinder.
How do I locate and access these or should this be a dealer fix?

Also exactly what mechanism happens when the tilt lock is engaged?

First check your battery. MP3's are very sensitive to voltage

The tilt lock is actually two mechanisms. Hydraulic to locks the shocks and mechanical to lock the tilt linkage. Sounds like your hydraulic system has issues since it is staying upright but getting squishy. Check the fluid level. It is just brake fluid. Sometimes topping it off will allow it to work again for awhile anyway. The bad news is you have to take off all of the front plastics to get to it to work on. If you have a good shop nearby I would just take it there.

Here are the beep codes

You can also just pull the horn relay from its socket, empty the reservoir (and/or remove the "rods" from the shocks), and call it a day.

I disabled the hydraulic part of the tilt lock system on my mp3 a few months ago because it was starting to fail just like yours (different cause probably, but same symptoms anyway).

Read: https://modernvespa.com/forum/topic175187

Can't say I regret that easy fix. I even like it better this way