Should I update my Bontrager Litespeed mountain bike?
I'm not sure why you think you need disc brakes on a hybrid, but if you want a hybrid with disc brakes it might be cheaper to buy a newer hybrid with that comes with them (for same quality brakes). You won't be messing around with questionable adapters. In fact you might not even be able to source an adapter that will work with your fork depending on the fork (disc might not fit in there, adapter might not be able to attach to the particular fork).

If you decide to replace the fork you will probably have a hard time finding a straight steerer fork with disc mounts that doesn't have a significantly bigger A to C measurement. A larger A to C will affect the geometry: slacken seat and head angle, raise bottom bracket etc. The slacker head tube angle might be fine (90s bikes were steep) but the slacker seat tube might be too slack resulting in a poor pedaling position and a hard time keeping the front down on climbs because your hips end up behind the rear axle shifting your center of weight too far back.

Your bike seems like a good candidate for a restoration . Stick with the v-brakes (new pads). Do a full tear-down and lube. Clean everything up. If it has an old elastomer fork that is shot you might be better off replacing it with a rigid fork, like a Kona Project 2 from that era. Grab some street friendly tires (the old tires will be shot from age regardless of remaining tread). Check the drivetrain and replace parts as needed (possibly cassette, chainrings and chain). 7 speed Shimano parts should be easy to source and inexpensive (safe bet it's running a 3x7 drivetrain). Might have to settle for it saying Alivio instead of XT or XTR if you want to buy new, but you can find compatible parts new.