Can I put a plastic Mat Under Washer?

The office chair mats come in two types. One with spikes for carpets, which are obviously inappropriate. The other type for hard floors is quite smooth underneath. The ones I've used move a little on the floor. I would imagine with a spinning washing machine on top they would be constantly sliding around. As long as they are secured in a specific space or otherwise held in place you could find it a nuisance. There is not problem as far as the washer is concerned though, as long as you can stop it sliding around it should be fine.

Alternatively you could consider some thing office floor tiles, though you could have a similar issue with them moving on spin.

Specially designed washing machine pads

There are plenty of feet for washing machines designed to cut down on vibration and consist of rubber or plastic cups that the washing machine feet slip into. You might want to try some of them. Search Google for - washing machine feet vibration