You could look at 12" deep kitchen wall cabinets but don't know that you could get drawers. Other than than, look in used furniture stores, CL, thrift stores? Good luck - 10" isn't much, even for a bookcase with doors or a nighhtstand. Does she want it permanent or freestanding?

I have the same situation. You might look in the Home Decorator's catalog online or the JC Penney site. I remember them both having cabinets that were small in width and depth. You don't by any chance have a corner do you? There are a large selection of corner units.

You may want to think out of the box .. and use something that would not be a normally used for storage.. my mind is blank right at the present.. but think about it.. you can use baskets,over the toilet cabinets or shelves, I have even seen a shelf over the sink it would be narrow and not very tall but could hold some makeup and maybe a jar with combs and brushes in it.. have you ever seen a table in a foyer that has a mirror above it.. maybe something like that as they are usually pretty narrow.. and some have doors on them.. infact I have this in one of my bathrooms.. and it has shelves and two doors on it.. not very deep but has space to sit a basket or vase on it.. and I have a mirror above it..
just some thoughts..

Maybe check the unfinished furniture stores around you for a bookcase that can have doors made for it.

Or check Ikea. You would have to bring your tape measure with you but they have a lot of clever design ideas.

Is there a possibility of taking out the sheetrock and using the recesses between the wall studs for a little extra depth? The wall studs can be trimmed out to look like dividers when the doors are open.

Here is a link that might be useful: Unfinished Furniture

I solved this by using a CD cabinet. They come in all sorts of styles and price points. You can get them with solid doors or glass doors.

Here is a link that might be useful: CD Cab

A surface mounted medicine cabinet. We bought one for our old bathroom. It was wide, about 5 inches deep, and had two doors

Why not use one of those vinyl back-of-the-door plastic shoe holders that have all the pockets on it? They have so many pockets that everything can be neatly organized.

Same problem in my daughter's very small bath. She hung two ikea bars- designed for kitchen storage- on the wall. She then hung containers on them. Hers look like tin flower pots which slip into a circle that hooks over the bar. Each container holds a different type of makeup, hair product etc. and the tin can be lifted off and brought over to the sink for use. We also installed three 8" shelves over the toilet for towels and baskets for bulkier storage.
I also agree about shoe holders. The inside of every closet door in my home has a clear shoe holder and none have shoes! Hats and gloves downstairs, sewing things in the den, first aide stuff, extra soap shoe polish etc in the linen closet- you get the idea. Makes little or odd things much easier to find.

Since you are asking about getting a floor cabinet for the bathroom...........Target sells items in smaller sizes for the apartment or small home dweller. This one below is 10Dx11W. The have other styles that are wider too.

Here is a link that might be useful: Target Cabinet

There are quite a few choices at IKEA, both floor cabinets and wall cabinets that are 12" or less in depth.