I'd like to build a couple custom racks out of stainless (don't want to paint or powdercoat) and the local maker space has both tig and oxy-acetylene equipment. Which of these methods would be best? I assume if brazing I'll need some kind of fancy filler with silver?

I would suggest going with Fillet Pro (and the appropriate flux) from Cycle Design for brazing.

Stainless Filler wire is commonly used in Frame building.. TIG.

I braze stainless racks with nickle silver. It's really a bronze-nickle alloy. You'll have to practice with it. It's actually a little more forgiving than stainless with silver in my experience. One issue is that you can build up a sensitivity, so good ventilation is key.