Can I get a 24" wide drop-in sink and mount it directly onto a 24" cabinet base? I assume that I will need to custom build the cabinet to basically frame the sink. If I do this, are there any unforeseen issues?

With a custom built cabinet, I don't see any reason why not.

Issues--

No room/ no way inside the cabinet for mechanical fasteners to mount the sink to the cabinet. But there are excellent adhesive products like Lexel, etc. that would glue & seal the sink rim to the cabinet.

Very little/ no margin for error as regards finished cabinet dimension and dimensions of sink rim.

Give thought to the deck area behind the faucet. Will you need some space behind the faucet? Also, the faucet choice--don't want to bust your knuckles against the wall operating the faucet. How to protect the wall?

Also, keep in mind that since the front of the sink rim will be flush with the front of the cabinet, that means there will be no counter top overhang to protect the front of the cabinet/ cabinet doors. To look right, the doors will need to be inset in the frame (not overlay doors). Inset doors require more skill to fit & hang. And there is still the potential for water to collect on top of the doors--not good.

Alternatively, you could design an inset into the cabinet itself. A 'stepped' front where the apron of the front overhangs the doors. (And think about the door pull, too.) This would be the better choice although it reduces interior storage space.)

I have a similar situation of needing to replace an ugly, old, cheap, pitted, stained laundry sink with very little extra room. I'm considering getting a commercial (single well stainless steel restaurant) sink. You might consider that as well, if you can live with the industrial look of it. IMO, this would be the easiest & cheapest solution.