Does pickling wood with steel wool and vinegar harm the wood?
This treatment only affects the surface of the wood.  It's not going to weaken the table.

That said, the effect of a vinegar/steel wood stain is going to vary a lot depending on the type of wood you're staining.  It will be best to test your entire finishing process including the clear finish on the actual table.  You can use the bottom or some other inconspicuous place to experiment.