I've run a few training runs in the rain, and I ran the 2014 Wine & Dine half...it rained pretty steadily the entire time.*
Honestly, it's not the end of the world. It's not IDEAL, but it's not terrible.
Best advice I can give you...
1. Wear a hat. SO helpful for keeping the rain off of your face. I wear hats for my rainy training runs and I didn't have one for Win & Dine. I think that would have helped a lot.
2. Wear good moisture-wicking clothes. My running skirt was perfect, but my shirt did nothing to help wick the water. It just felt heavy very quickly.
3. Bring something to keep you dry while you are waiting for the race to start. I'm not sure if you've run a Disney race before, but you'll have some waiting around to do. For Wine & Dine, we bought ponchos in the parks ahead of time and wore them while we were waiting in the corrals. Once it was our corral's turn to start, we took the ponchos off and tossed them in a nearby garbage can. There's no way I could have run in one - it would have been way too hot. You may also want to bring garbage bags to sit on during the wait so your butts stay dry.

4. If it does rain and your shoes get soaked, you can stuff them with newspaper afterward - this will help them dry more quickly than just having them sit out.
Remove the laces and the insoles...
- Stuff shoes with newspaper and leave it in for an hour
- Remove old paper and stuff shoes with new newspaper...let sit for 2-3 hours
- Remove old paper and stuff shoes with new newspaper...let sit overnight
5. Bring an old point & shoot camera to use for on-course photos - I mostly use my phone for pictures but I don't have a water-proof case for it. Having my old point & shoot at Wine & Dine was a lifesaver. I still got pictures and my phone didn't get soaked.
Biggest piece of advice - if it actually DOES rain, accept that it's happening and don't freak out. Make the best of it. You're still running in
Disneyland!!
*I still feel like a total bad*** for completing that race without too much whining. 