There may (or likely is) some MMS gateway that will allow your app to appear as another phone number (or short code), but I'm pretty sure you can just have people send the MMS message to an e-mail address. I believe that's standard (as in all carriers support it).
You'll just have to parse the sender's e-mail to find the phone number, and grab the attachment to find the file. You may need to 'detect' the correct attachment as some carriers add branding elements to the email.
Updated: From Wikipedia, it seems MMS to email gateways are 'common', but I guess not required.
E-mail and web-based gateways to the MMS (and SMS) system are common. On the reception side, the content servers can typically receive service requests both from WAP and normal HTTP browsers, so delivery via the web is simple. For sending from external sources to handsets, most carriers allow MIME encoded message to be sent to the receiver's phone number with a special domain.
So if your app needs to be completely accessible to everyone, this solution may not work. It is worth mentioning that popular file sharing site drop.io just uses an e-mail address for its MMS send/receive.