How can I turn a vanity into a small dresser?

I think this discussion will give you some helpful direction: http://www.hometalk.com/diy/paint/q-i-have-an-art-deco-bedroom-vanity-i-want-to-repurpose-1342464

And here are some great Dresser update ideas: http://www.hometalk.com/categories/woodworking/furniture/dressers

https://www.pinterest.com/explore/dresser-bathroom-vanities/

There is no picture so I am going to just toss out an idea.Get some of those fabric storage boxes at any Home Depot or online.They come in almost any color and several sizes.You could use them as drawers.Martha Stewart makes some.They are about 5 bucks each.

Are we talking about an antique vanity table or a buildergrade bathroom vanity?

Assuming we are talking about a bathroom vanity - you can buy drawer kits from any good hardware store, and custom build your drawers. You can use false fronts from inexpensive vanities to be the drawer front. If you have space for shelves inside doors, by all means used boxes or baskets to hold possessions. For a top - wow, endless choices. Check out the ReStore in your area for a table top that could be cut to size, a nice solid old door, or get a right-sized piece of tempered glass to put over a shadow box with mementos or scrollwork, even fabric for a changable look. If you get lucky, a miss-cut piece at a local stone cutter could be a bargain price and would look really great. Post photos when you get your project done!