Yes, a Jennings J22 will fire without a magazine inserted.
These retail for about $85 new.
That's a task best left to a gunsmith.
.22 LR rim fire
The thing that keeps the slide on a j22 frame that won't stay on is the slide catch. It is designed so that the slide mechanism can be locked in place without risk of it moving.
frame
The estimated value of a Jennings model J-22 pistol is between $80-$150 dollars depending on finish and condition. They are becoming more rare and are going up in value rapidly. They are a part of the Ring of Fire pistols and have a unique place in history as the original "Saturday Night Special", which may ultimately make them a must-have with collectors or "mouseguns".
Best thing is to have a gunsmith show you.
Recommend you ask a gunsmith for help
Try e-gunparts.com Better yet, search inner city alleyway dumpsters on Saturday nights or police evidence rooms in the 1970s unsolved homicide sections or perhaps you could ask local crack dealers and muggers where they get theirs
Best to ask a gunsmith for help if you don't have a manual
They were about $60 new, and now about $100 in good condition at a gun show.
You will have to experiment with as many different brands as you can afford to find out which it likes.