The Edge 3 Stretto? That platform is rock solid. I don't think anything short of submersion will kill that power-base. The electrics (controls and power connections) are actually pretty simple and unlikely to be bothered by a little rain or splashing. That said... being out in a full downpour is a little different. I've never cracked open the controls so I don't know if any actual 'weatherproofing' is built into them other than the snap or screw enclosure.
So... there's two areas that are most exposed, the controls themselves (joystick and speed control) because even if sealed to some extent they are still right on top, and cord connections, things like the connection between the controls and the powerbase or between the batteries and the charger or control board.
If exposed wiring connections are not weatherproof, replacing them with waterproof connectors is super easy using something like
this.
If you're hesitant to cut up your very expensive wiring harnesses (don't blame you) you can waterproof existing junctions with something like
this.
And if that seems a little too clunky for you, one of my favorite products is a self-fusing silicone
'rescue tape' that works great for this job.
Now, for the controls and electronics boxes themselves, that rescue tape can be wrapped around the seems to good effect. Another technique I use is to give the electronics enclosures (inside and out) a heavy dose of NeverWet. It is a super-hydrophobic coating that was all the rage a few years ago. Some pretty cool
YouTube videos of NeverWet treated people getting disgusting stuff poured on them. The magic is that the coating dries very thin, like a coat of spray paint, so it can be applied right onto the circuit boards inside the electronics enclosures and then applied to the outside of the enclosure as well. Water, and almost everything else, will just fall off the treated area. It's cool but not too good to be true, it works like magic but the effect diminishes after a few months.
This is an interesting and common misconception about children and powered mobility devices. Guess what requires far more attention, fine motor skill, strength, and stamina than piloting a power chair? Walking. Now imagine telling an able bodied three year old that it's just easier for everyone if he'd sit in a stroller all day and get pushed around.
More than that... one of the truely special aspects of this particular chair, and I suspect a reason the OP had in choosing it, is that it's designed to lift the occupant to eye level. In this case, probably eye level of a 3- year old, but the point is that it's designed to mitigate the invisibility disease that so many in seated mobility device seem to catch when out in public.
In my experience, I've seen 3 year olds manage their own mobility even without legs so I would not presume to suggest that another must be incapable of piloting a powerchair.