I could have sworn that from the factory, the 94-01 Integra had a cable-operated remote trunk release.

All you had to do was lift up on a small lever near the drivers seat, and the cable would trip a lever, unlocking the trunk.

Now, if someone installed an electric solenoid as part of a keyless entry system, then it's certainly possible that the solenoid itself has failed. Also, the cable may have snapped or disengaged from the lock.

Open the hatch from the outside using your key, then remove the trim panel that covers the back of the trunk. The lock is bolted to the inside - obviously - and you'll be able to visually inspect the lock and related components.