I'm an ophthalmologist and also an RGP wearer.
If your RGP's are fit properly, it's very rare for them to pop out from spinning rides, roller coasters, or even from wind.
HOWEVER, you say that you were recently "diagnosed with an eye condition" requiring RGP's. Since it was a recent diagnosis, I'll assume it probably WASN'T regular astigmatism (which is what I--and probably the others on this thread--have.) Regular astigmatism is a dime a dozen and can typically be corrected with spectacles and gas permeable lenses (RGP's) but NOT soft contacts. It's by far the most common reason to need RGP's.
But it's not usually a "recent" diagnosis. That makes me wonder if you have keratoconus--a condition where the cornea becomes cone-shaped. This is a form of IRREGULAR astigmatism. RGP's work great for keratoconus, but they become a medical necessity, as spectacles can't correct IRREGULAR astigmatism.
With irregular astigmatism, the RGP lens fits, but not as perfectly as it might for other patients. THIS MEANS the contact lens is at higher risk of popping off.
WHEN DO RGP's POP OUT?
1. Poor fit, whether it's your cornea's fault, or the lens'.
2. Sudden, tight blinking--as when a foreign body hits your cornea or conjunctiva. As mentioned earlier, you can minimize this risk by keeping the eye WET WET WET. Use artificial tears. "Rewetting drops" are a bit of a scam for most people. Cheap artificial tears are fine if you can tolerate them. The more expensive ones have better (or no) preservatives. I USUALLY tell patients to buy the cheapest artificial tear that they can tolerate.
3. The biggest risk at WDW is almost certainly jumping into swimming pools and aggressive water slides. Patients often lose contacts when jumping into a pool without goggles. (Routine swimming won't do it.) The only time in ten years that I've lost my own contacts is from diving into a pool.
In fact, on our last trip, this is why I couldn't go down Summit Plummet at Blizzard Beach. I wanted to wear my goggles to protect my RGP's, but the CM wouldn't let me. I had no place to put my contacts, so I had to wuss out.
Bottom line, bring some artificial tears and an emergency contact lens case.
And if your eye doctor disagrees with any of my advice, just tell him you read it on the Disney web site from some guy on the internet who said he was an ophthalmologist. We always love to hear that kind of stuff from patients.