Hi Elisabeth,
Oh sorry I misunderstood your requirement in the first place. Forget about what I have pointed out earlier. The HTTP Access to vSphere Server Files is meant to access datastores etc.
Currently I am not aware of any vSphere API which will help you with your requirement. I just looked into my vCenter 4.1 and found there a Tomcat 6 server. So far I have not played around with the vCenter Tomcat but you could try to write a simple webapp hosting a single Servlet which implements your upload functionality. Then install that webapp into your vCenter Tomcat. I think this could work.
Good luck :-)