You could probably go into any store in any of the parks and ask them to call your resort for this purpose. I have had to call the resort from a park when we reached the limit on our room key/charge card. I popped into a store in Magic Kingdom and the CM had no problem with calling the resort for me. I am sure they would do the same for checking on room availability.
I also remember once that I was having lunch at the Big River Brewery at the Boardwalk and I needed to call the Animal Kingdom Lodge to check on the room. I was able to use a Disney phone to make the call without having to pay for it. However, don't know exactly where you would find similar phones in the parks.