Find a good quality marine sealer - one that's used on boats. It should be waterproof enough for your vanity. I'd think you'd need to apply several coats.
Buy a clear shower curtain. Place curtain over fabric or paper to be laminated. Cover this with wax paper. Iron on high heat until fused. Peel off wax paper.
As An alternative you could check flooring scraps of oilskin or such. We covered a beat up wooden desk with a piece in a big black and white squares pattern. It looks fun and is functional with the grid for a work surface. You could easily match or complement your floor or walls.
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