Hi. That is a very distinctive looking unit, i understand why you want to keep it. This one use a motor up down for the top lid and motor for the CD tray but the Deck seem to open with the release of energy accumulated in them when manually pressed down. Usually this is a spring but perhaps it is a small air piston in that one that would need to be repositioned up or a spring that lost vigor overtime (or a plastic one that snapped by drying). The opening release of that does not feel mechanical, it seem to be a button that activate a release mechanism (that may be a small motor or a simple solenoid).
Cassette deck can eventually have rubber strap melting over time (the latex bio degrade into a goo) but rarely the opening mechanism fail. I would suspect perhaps dust or oxidation on the button contact if they rarely were used. A blown motor or solenoid is unlikely to cause both to not open. It might actually open but dust prevent the movement or keep those lids to come up being sticked.
That is the kind of job i think any electronic repair shop should be able to perform with good disassembly data. So i don't think any part will need to be ordered but only some cleaning of the internal area.
Here is the service manual to help in that task.:
The closest place to service it (if you don't want to attempt it yourself) is IT Electronics:
I hope this answer you to your satisfaction and that you will be able to use it again soon.