Hello! The keys may or may not move, depending on the design of the
piano. What indicates to me that yours should not, is the lack of the
key lock lever.
You could check by removing a key and having a look if any lead weights
are perhaps missing in the key front. If the leads are in the rear
half of the key, then everything is as it should be and the only way
to make them move is by changing the key weight from being rear-heavy
to being front-heavy. This will upset the balance of the piano action,
though, and I do not recommend it.
I have found mostly the English player pianos to have non-moving keys.