An example of "Waterfall" furniture of the depression era.

Yep, what Sam said. Art deco. They were affordable and mass produced but nice. Unfortunately (unless you're shopping!) they seem to be sold for almost nothing now.

I love them. Looks like the veneer is in good shape on yours--you just need a few handles. Are they bakelite?

1930s-1940s "waterfall" style with walnut veneer. My parents had one almost identical, bought new right after WWII. (1946 or 47)

Should have had a large circular mirror attached.

Nice vanity, but it's going to be hard to replace the handles.

If it has a bench with it, you may find the manufacturer tag underneath the seat. My mom's vanity was part of a 5 piece bedroom set and the only id was under the bench. But they were mass produced and I'm not sure knowing the manufacturer and date will alter the value much.