Please don't worry - breakdowns (and short-life batteries) are the exception, not the rule.
As I said above: Make sure you put your Rental vendors contact information (including office hours and after-hours emergency contact info) into the "Contacts" or "Address Book" on your phone. If you do have a problem, you will have the information you need at your fingertips. (literally!)
If something were to happen - let's say the rental scooter just croaks - you call the Vendor (if it is during their regular business hours) or you follow their after-hours procedure. If for some reason you can't get ahold of them, then reach out to the nearest CM (Cast Member) - regardless of what position they hold (cashier, wait staff, maintenance, attractions, doesn't matter) and tell them you need assistance.
Depending on what your exact problem is, they may tell you to go to Guest Relations, or they may be able to dispatch some other form of help. The main take-away from this is: Reach out to a CM as soon as you know you need help. If the first one you talk to doesn't know what to do, or seems unsure of how to help, find another CM. WDW is crawling with them. Be clear and concise in describing what you *need* "I need help because my rental scooter has broken down, and I can't walk all the way back to the buses at the front".
But the chances are *very good* that you won't have any problems. Rent from a reputable vendor. Make sure you have all of their contact and emergency information in your phone. Try to meet the vendor upon delivery, so that they can go over all of the features of the
ECV with you. Know if there is a parking brake, or a brake release, where that is. Charge the battery EVERY NIGHT and make sure it is fully charged before leaving the Resort in the morning. Stop as soon as you notice a problem with the ECV and call the vendor - don't go all day thinking it will "heal" itself and then find yourself without wheels after they have closed for the day.
You'll be fine. Seriously.
Have a lovely trip!