I've tried it on 2 different phones which should support it. Both fail to load the URL.

I've also tried typing in my phone number so chess.com can send the app to me over SMS but that returns an error message on the website, it randomly chooses one of these 2 errors "Authentication Error" or "Failed to send a message to your phone"

Try to point your phone(s) directly to:

http://www.chess.com/mobile/bin/www.chess.com/Chess.com.jad

or

If you have a computer and a phone link maybe you can DL the mobile app to the computer and transfer it to the phone?