Yes you can use screws and washers. If it's wood or a brittle plastic then you may be better off pre-drilling undersized holes into it to lessen the chance of it cracking, and as Bartman mentioned, use screws only as long as needed to prevent poking through when the seat padding is compressed from sitting on it.
Also over a few years time and use, vinyl can tend to shrink and rip out at the fasteners so I would think about tacking down the perimeter with some kind of adhesive in addition to the screws, a good volume of caulking or construction adhesive etc, should work. On the other hand if it is prevented from shrinking it may just crack instead.