SMS (and MMS) for your phone depend on your phone's SIM card, and your mobile provider's service, the one designated by your SIM. This is not VoIP.
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.