Worldpainter worlds will not automatically create caves when exported. Vanilla caves are a step in the terrain-generation, that is skipped for worldpainter worlds.

Therefore, you will have to create caves yourself. Worldpainter provides 3 out-of-the-box solutions, that are very easy to use: caves, caverns and chasms layers.
If you want more control over the caves, you can make your own custom cave layers. There are guides on using cave layers here:

simulating vanilla-styled caves

creating custom cave layers