Welcome to the community!
Unfortunately, I don't think there is a way for mobile providers to allow online access to SMS. I think the reason might be that the systems for SMS/MMS and internet access are segregated from each other. However, if you have an iPhone, you might be able to do so via iCloud (see here). If you have an Android device and use Google messages, you might also be able to achieve something similar (see here). There may also be some other apps which offer that feature (see here).
Hope this helps 😀