So first things first: if it’s not fun, don’t go. If it’s pouring down rain, if it’s cold and deep mud, if the kid’s-and-pony aren’t going to enjoy it, there’s always another event. Seriously, talk with kid’s coach/trainer and don’t over think it.
That being said, I’ve had some of my best events in the rain. Most horses do just fine in it and couldn’t care less, and there’s always the scrappy pony who thinks making the biggest splash possible in the puddles is the Best Thing Ever. Some concrete suggestions for your questions:
Yes, you can wrap stirrups in vetrap, though depending on what kind of inserts she has in her stirrups already, she may have a non-slip surface there. I like these (http://www.doversaddlery.com/super-comfort-pads/p/X1-0752/ ) which wrap around your existing stirrup and zip-tie into place; others like the sandpaper or metal inserts (more like these: http://www.doversaddlery.com/supergrip-stirrup-pads/p/X1-0737/). I leave these pads on my stirrups all the time - they’re fine for dressage and jumping.
No need to buy new reins, just tie a knot in the end of hers. Will give her something to hang onto and will shorten them appropriately.
Jackets: I like a fitted rain shell that isn’t too noisy - think Patagonia style rather than big-and-crackly. Have the kid ride in it once or twice beforehand if she’s not already so she’s comfortable.
Gloves: I actually sometimes go bare hands in the rein as I feel like I have better purchase on the reins, but there are gloves that have either a tacky or a pebbly grip that some folks like. I find pure leather gloves to be way too slippery but most of the fabric ones are fine. The knot on the end of her reins is going to help her here too - keeps her from losing them even if her hands slip.
How to see in the rain? Well, if it’s raining so hard that she can’t see, suspect they’ll either suspend competition or you/her coach won’t send her out in it! Most of the time it’s really not an issue. If she wears glasses normally, she may want to think about some of the wipes you can get that can keep them from fogging up.
Staying warm afterwards? Throw a cooler or rain sheet over the pony and put it back on the trailer or in the stall to tack/untack. Throw a fleece over the kid. Apply hot chocolate to kid; hot coffee to adults. Repeat as necessary.
You’ll be great. She’ll be awesome. Have FUN.