Hi Bryan,
The short answer is yes, its capable. Both of our hosting partners, Insyq and Right Networks, have managed to get the Method sync engine to work with their hosting solutions. I believe there are others that have been able to get it to work as well. With that said there are some considerations to take into account. Depending on the setup, there is likely some work needed to be done to get the proper permissions set on the server in order for the sync engine to run properly. A cloud hosting solution will likely run with Terminal services or Citrix, and those can present a number of complications. I'd recommend checking out this Help Center article, Server Installations, and also this forum post for more info.
Hope this helps,
Adam