Common approaches for thin stock:
-hold by the edges (clamps, screws, whatever) with a sacrificial subplate. 3/4" MDF is common, plywood or scrap hardwood works. Don't forget that one can always drill small (1/4") holes in thin stock where there are large expanses of unmachined area to allow screws or bolts to be used as hold-downs, with washers as needed, on a subplate. That's not cosmetically elegant- but it works just fine, and allows machining the edges as well.
-an alternative is to sandwich the thin stock between pieces of MDF. No clamps on the stock, just clamp the MDF well. If you have features that can all be done at once (several internal holes), that works fine. Then re-mount on a single piece, re-zero (use one of the holes) and then cut the edge profile. Registration will only be as good as your set-ups, but usually adequate. This approach also helps with grabby drills.
- use doublestick tape to hold (3M 95395792) on a subplate. Other doublestick tapes sometimes work under light forces; this one is pressure activated, so use a heavy clamping force (press, vise) to push it down hard. Still don't want to take deep cuts. Best with metal subplates.
-with a metal subplate, cyanoacrylate can be used. Work and metal subplate must be clean, flat. Machining forces may be too heavy, especially for heavy cuts. CA adhsives can be readily broken by heating to about 300C. Do that outside, avoid fumes.
-Mitee-bite edge clamps (again, sacrificial plate) or the round screw-down ones in a vise jaw. Issue here is that you're drilling a rather thin plate, so any size beyond maybe 2x hole size is going to be a bit iffy (think potential frisbee with sharp edges). Edge clamps will allow heavy cuts if used properly- Mitee-bite claims 2 tons of clamp force or more.
Personally, I'd never try to drill thin stock <1/4" held in a vise the way you seem to imagine doing it (between jaws, maybe up on parallels). Mostly, it'll work. The time it doesn't, if the stock gets loose it's a flying knife edge. 3/16 is at an awkward point- not really thin, but not really thick enough for a vise. Subplate and bolts would be my choice.
For larger diameter holes, pocket it if you have CNC capability. That avoids grabby drills. ("Larger" = >1/4", if you have CNC).
