Disney's Hotel Santa Fe has pay-per-minute internet kiosks in its first-floor Observatory. Climb the steps between the entrance to La Cantina and Rio Grande Bar to find it. (info from dlpmagic website)
Hotel Cheyenne has pay-per-minute pc's on the ground floor just off reception follow the signs for the toilets. I'm pretty sure these took 2 euro coins, I used them during our October 2009 trip which coincided with the fuel shortages and meant I had to keep checking our flight home was still on.
Have read recently that there is free wifi inside the train station but not had chance to check it out yet.