You may find what you need in the catalog of a semi-custom kitchen cabinet manufacturer. They usually have some 18" depth cabinets or you can order a 21" with a reduced depth option.

The only recommendation I have is to key in "prefab vanity 18x47". This is a custom size though because the standard size is 21 so you may have difficulty finding one within your budget. Good luck!

Bertch (Bertch.com) makes an assortment of cabinets and offers an 18 inch depth. My neighbors bought one and are very happy with it.

Be aware that you'll have to get a smaller sink than you might imagine. I'm also getting a a shallow vanity (but having it custom made) and it took a little bit of searching to find a sink. Of course, if you're getting a vanity and sink/counter together, you won't have a problem.

This one from IKEA is the size you want but has two sinks: http://www.ikea.com/us/en/catalog/products/00293668/. We have two of the smaller 39" ones side by side in our master bath in gray.