It's possible to do the run without using the motorway, but it does take you through a few towns/slow speed limits and if you're just following the sat nav, yes, you'll do a little autobahn.
The vignette costs so little and it's so easy to buy that it's pointless NOT buying one. If you're going over Fernpass (the way most people head from Garmisch to Stelvio) the first petrol station over the Austrian border sells them. It takes five minutes to stop, top up the tank, and have the flexibility to legally use the autobahn if you want/need to.
The vignette just gives you options. My preferred option being avoiding Fernpass on the way back as it's always busy.
ETA: a route explanation as that final sentence is pretty meaningless without it...
I'd go from Garmisch to Imst (Fernpass), autobahn to Landeck, through the tunnel, into Switzerland to Zermez, then through the (toll) tunnel to Livigno and round to the foot of Stelvio outside Bormio (
like this). After crossing Stelvio it's back to Garmisch via Telfs and Mittenwald – either directly (
like this) or if you like high-mile days, over Timmlesjoch