is 3/4 mdf strong enough to build a cabinet carcass for my shop tools
such as a rolling cabinet for my bench top drill press; mortiser; etc?

Would it be better to use 3/4" hardwood plywood or a dimensional
lumber frame with mdf skin?

I doubt mdf on it's own would hold up. Depending on the weight of your
press, plywood alone might not do it either. I'd suggest a 2x4 frame with a
couple of extra cross members to support the press, covered it with mdf.
That's what I did. It's quite sturdy and if we ever have an earthquake
you'll find me hiding underneath.