You really aren't ever that far from some kind of cover or shelter - your best bet in a major downpour is to cover yourself and the ECV with a couple of ponchos and head for whatever's closest to wait it out. Heavy rain can cause standing water that can splash up underneath the ECV that ponchos won't protect against, so driving around is risky.
In a milder rain, you can get away with driving around while covered in ponchos. I usually still head for an indoor show or attraction, though. No need to tempt fate.
Both rental companies I used provided a plastic shower cap to put over the controls to keep them dry. Wearing a poncho and letting it spread out around me over as much of the ECV as possible kept a lot of the rest reasonably dry. I brought along an extra poncho to cover the whole thing if I ever needed to leave it out in the rain. (I've heard other people say they brought a shower curtain or plastic tablecloth.)
We were stuck in the disabled area watching the parade when a downpour happened, there was no way to get out. I covered myself and the scooter in a poncho and hoped for the best! Sometimes there's no avoiding it.
We got caught in World Showcase when the heavens opened - and it was an absolute onslaught which lasted for hours. There was no way we could just sit in the Chinese pavilion and wait it out: we had an ADR at Coral Reef to get to apart from anything else.
I put a plastic bag over the console and I was wearing a poncho, but I unfortunately trundled through a 1" deep mini lake and that was what killed the scooter, we think. Shame they hadn't told me about the reset button!!!
I'm seriously considering buying a full custom-made scooter poncho for next year: it covers you and the entire scooter (the front panel is see-trough plastic so you can see the console).
if you get caught in rain, remember to make sure your poncho covers the back of seat so rain does not come down between you and seat. when covering when parked put seat back down so no puddle in middle of cover you go to remove cover. shower cap over controls works great and learn where the reset switch is. only problem I had was once when we waited out rain and had to call to learn how to restart. they were able to do over phone but it did cost some time
I was assured more than once that the battery connections were safe from water ingress and that it wouldn't be a problem. However, the scooters were in poor condition and the battery covers were both held on with adhesive Velcro and 'a wing and a prayer' as we say over here!
The problem with wearing an ordinary Disney poncho was that my arms obviously stuck out as there aren't any sleeves, so the sleeves to my fake sheepskin Tinkerbell fleece got drenched with cold water. As I can buy a proper full-cover rain poncho for less than £20 (tax free as you can buy under a disability VAT exclusion), it's a bit of a no brainer. It will do up to the neck with a zip, and if I'm careful I'll be able to extricate myself from it leaving it fully covering the scooter if I have to leave it out in the open while I go in somewhere.
http://www.completecareshop.co.uk/wheelchairs/scooter-clothing-and-covers/scooter-cape/
I'm sure you can buy something very similar in the US - and possibly cheaper!
I use the same method at WDW that I use at home on my own scooter: I take a large garbage bag everywhere with me, fits over the driving stem completely and still allows me to use the controls. I used this method in Oct at the food and wine festival in an hours long downpour and never had a problem with the scooter continuing to run.