Yes. I've done it several times. A lot depends on the type of chair. Usually, you can run a blunt-ended table knife round between the plastic and the seat pad and it will pop out. You can then just use a staple gun to attach the new fabric. You might have to unscrew the back pad. I used the staple gun to cover the front of the pad, then applied piping round the edge. I hand sewed the back covering on using a curved upholstery needle.
The last one I did though was not so straightforward. It's a more modern high backed type. The seat has to be unscrewed and the plastic underside just drops down. On mine, all the screws are different so need to note which goes where. Not easy to staple into plastic and I ended up glueing. The back partially popped out and I used glue again.
Somewhere I have some pictures. Will post if I can find them.