Question: Is there any way to fire off a data subscription from using the SSRS web services?

Answer: 
Unfortunately the SSRS web services do not support running data subscriptions. If you want to run subscriptions through a web service call, you could expose a web service on a server that runs a sp\_start\_job command to the SQL Server Agent job for the associated schedule. Sounds dangerous to me, but it is feasible.
Sure looks like it's possible:

<http://geekswithblogs.net/shervin/archive/2007/06/20/113351.aspx>