You'd be able to get all of GT Sport's single-player campaign mode stuff since the updates are handled through the Playstation Store servers, but assuming they don't do an update to address this, you wouldn't be able to save any progress. The way it currently works if you try to play it without ever connecting to the GT Sport servers is that every time you launch the game, it's like you're playing it for the very first time.
And it's worth pointing out that the arcade mode would also be extremely limited, since most of the tracks are locked until you reach certain levels... and while you can technically receive experience and level up offline (at least temporarily), in my testing doing so won't actually unlock anything in arcade mode unless you're connected to the servers. So effectively you'd be limited to Northern Isle Speedway, Dragon Trail, and Colorado Springs for arcade mode.
While I do enjoy GT arcade modes having unlockable content, I'd prefer that GT7 make the content unlock through actually playing arcade mode itself rather than relying on progression via other modes... either that, or simply not have locked content for arcade mode. Definitely don't repeat GTS's mistake of making it dependent on progression that can't be achieved if the servers are taken down (again, assuming the games wouldn't be updated to fix this on account of, y'know, capitalism).