After hiring a scooter from a bad place the first time - which died halfway round the World Showcase when the heavens opened - and that was after putting a bag over the tiller - I've been very wary of driving in rain.
The people I hire from now (Florida Mobility Rentals) told me not to worry about rain unless it's very heavy. In fact, he drove it to our accommodation in drizzle without even putting a cover over the tiller.
Last year OH and I got stuck at MK during the worst rain we've ever encountered (the tail end of Gaston and the run-up to Hermine). I do have a full scooter and body cover but it hasn't been out of the bag yet. I flapped my poncho front over the handlebars and operated the scooter from inside so my arms also stayed dry. We were watching the ground for deep puddles (we were stuck by Guest Services and there was a worryingly deep puddle right at the foot of the ramp), but made a run for it. Well, I made a slow roll for it - I didn't want to kick up a shock wave as that could have splashed the battery and cut the motor out.
So, yeah, cover the tiller, try and avoid puddles, and if you really can't, go slowly through them.