Is it really just getting my texts from my phone? Are the phone and the web app not pulling and pushing from and to some external service? Is this a real-world constraint that is a problem for all apps that want to act as an interface for SMS?
SMS isn't like email, where there are MTAs and MUAs. SMS is very peer-to-peer. SMS messages go through the mobile network and are delivered directly to your phone with best effort. They aren't stored on a server. MMS stuff like images can be on a server, so you receive your notification and then you hit a "Download" button on it.
If you want a fully online service that doesn't (directly) depend on your SIM, then use Google Voice. You can obtain a Google Voice number if you have a phone. Then you can send and receive SMS, MMS, and voice calls over the web, on any computer or smartphone.