Seems to me even if it's a little stiff, it should move at least a little bit. Makes me think it's electrical in nature. Measure the voltage at the coil and make sure you're getting something.

Generally there is no need to replace a coil unless the old one fried, which is usually pretty obvious. Is the coil wrapper burnt looking? However, the sleeve could be replaced as these are a stand-in for more traditional lubricants on moving parts. I believe this assembly will be pretty standard, so I imagine they have the sleeve in stock? If not, try Pinball Life.

While you have the mechanism apart to replace the sleeve, look at the coil plunger/link. Perhaps consider ordering a new one at the same time as the sleeve. If the shaft itself is decent, make sure the end isn't mushroomed out. If it's basically okay, at least give it a good polishing. With the mech back together, it should move very smoothly.

If the mech doesn't move on its own even when properly rebuilt, most likely you're going to be looking for a broken wire. The coil is switched by a transistor on the driver board - but transistors typically fail "ON," meaning if the transistor failed, the rocket kicker would fire as soon as you turn on the game and never release until you shut it off.

There will be two wires attached to your rocket coil - or more likely, three, where one wire connects to one side and two wires both connect to the other side. The side with two wires is where the solenoid gets power. Check all the other coils and make sure the wire of that color hasn't broken off somewhere.
The side with one wire is the return path. It should make its way up to the backbox. Check to make sure it's not broken, pinched, or pulled out of a connector somewhere.