How can I play Arkham Horror remotely?
Short Answer: No?

Longer Answer: Not that I've used at least. The only Arkham-management-eqsue software I've used is the Arkham toolkit for iOS, but that only helps manage the location/OW decks and isn't really for distance play. 

That said, there is a program/system called VASSAL that people can make modules for that allow network play of various games. I've never used VASSAL myself, but I know that it can be used to play rather complicated competitive games online, and a quick search reveals that there is a VASSAL module for Arkham plus every expansion here that got updated a few weeks ago. I see a note on the page that all of the common item cards are blank to make sure people own a copy, so I don't know what other modifications if any the modules would need after downloading to get it into a completely easily playable state. That may be your best chance at a good online method of playing Arkham. If you give it a shot, reply with your experiences in some fashion; it would be useful info to have here.