If you have a serious issue, ongoing quarterly pest control, or aerosol "bombing" may be necessary.

Plywood or particle board underneath? 

It was a lot less work than I thought, but I did fill up several dumpsters with particle board and soggy drywall.

Check to see if it have particle board flooring which can give problems when it get wet.Joe Gore

@John Lindemann       If the house is safe (no fumes or particles), talk with the tenants.

My experience is that many of them are made of particle board and cheap materials.

Replacing particle wood under the kitchen sink that the tenant claims is moldy...

If there are particles of food in the sink water, they'll show up in the dishwasher too, and may cause a smell, especially after they sit for a while.If this is happening, the fix can be as simple as making sure the drain hose has that raised loop in it. 

You can always use particle board with handles screwed in, like the previous owner did for two duplexes I bought last year. 

Vanity all day for storage, and warping should not be an issue as long as you stay away from particle board