Walmart, Home Depot, Target... Or you can find some online.
Here's some on amazon I just found
https://www.amazon.com/dp/B002CGRVNE/ref=asc_df_B002CGRVNE5064722/?tag=hyprod-20&creative=394997&creativeASIN=B002CGRVNE&linkCode=df0&hvadid=167125445179&hvpos=1o7&hvnetw=g&hvrand=1929121719464190142&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9027958&hvtargid=pla-274449610711

Maybe cut a piece of non-staining rug pad - that is made to use for vinyl as well as other hard floors - to fit under each of the legs of the vanity?

I'm using felt backed sliders for under the living room furniture. They work really well, but some of it tends to slide when we sit in the furniture or the dogs jump up. So, one tip I read was to cut a piece of non-staining rug pad and place under the floor protectors used to keep the rug in place. It has helped for that and it is made with a material that would work with wet floors. If it were me, I might double up on the material as the floor protectors are so that you don't have dents as well as gouges in your flooring.

I'm not understanding why you feel you need pads as the reason for the pads is when you move furniture across the floor you don't want to scrape it and I'm assuming that you won't be moving the vanity?

If you need them, I've used these on my chairs with the wood flooring and perhaps you could replace the felt with rubber for better wear.

https://www.amazon.com/Anti-Sliding-Felt-Pad-Furniture-Alimitopia/dp/B01GOODUQG/ref=pd_lpo_vtph_60_bs_img_2?_encoding=UTF8&psc=1&refRID=1AAVJN9EKGHE8GHGW65V

All interesting ideas - thank you! I will look into them.

Hi, tedbixby. Thanks for replying. The reason is that the installation instructions for the flooring say to use floor protectors of at least 1" in diameter under any furniture legs. Our new vanity has legs, the whole cabinet doesn't sit directly on the floor. My concern is that felt pads might soak up water spills and hold them, and damage the legs over time. I'll check out your link - thanks again!

Hi, Lyndy, Thanks! I was aware that the protectors are to prevent indentations,, but I didn't know that rubber on vinyl might be a problem. We'll keep that in mind, definitely.