How can I use lace to cover a small vanity?

Yes, Mod Podge would work to glue it to the vanity with... Then water based poly would help fill in some of the grooves, but it'd take so many coats to completely fill it in. I think I'd get a piece of glass cut to lay on the top to make it totally smooth.

Not really, it would stay lumpy and if you filled it in, you'd loose the lace look. You can do the lace mod podge and get a piece of glass cut to go over the top. Sounds cute.

To avoid the lumpiness, you might want to use your lace as an overlay or stencil: https://www.home-dzine.co.za/crafts/craft-lace-drawers.htm

Instead of gluing the lace onto the vanity what about using the lace as a stencil? I used a lace tablecloth on my vinyl mini blinds, but the same process would work for a piece of furniture too.

https://colormethrifty.com/2018/10/mini-blind-makeover-with-spray-paint-and-lace.html

I love this!! Going to try it!!! Thank you sooooo much!