This suggestion comes from my experience with 740s.

Do you know if you have an actual key or a valet key?

Does the key you are using lock and unlock the glove box - if not, that is a valet key and it will not unlock the rear.

Try all the keys that you have - I think that the handles are larger on real keys.

See this url - https://volvoforums.com/forum/volvo-240-740-940-12/what-do-you-do-when-only-key-valet-key-25246/

Also, Volvo switches are a weak point - perhaps the switch has failed - try pulling the switch out and if it only has two wires, try touching the wires together to activate the trunk (sic) lock. All a switch with two wires does is to connect the two wires.

A lot of Volvo procedures start with a small screw driver to pry gently. The difficulty is to figure where to pry!

Many things snap into place, one side has a hook, and the other a detent - if you pry on the detent side, the item comes out, if you pry on the hook side, you break the hook. Try prying gently on each side, with luck you can figure out the correct side. If you have no luck, try the top and bottom.

If the switch has more than two wires, you need to try another switch.