is there any fee to trapping bobcats besides the trap?

No not really. You pretty much just need to a kanabear $ wise. The rest is what you can salvage and put together for a cubby, then you need the bait.

Check with your state wildlife agency most states have a trapping license and regulations. In VA you need a license to trap unless it is your own property.

What Criggster said. Most all states require a fur bearer license of some sort that is separate from a hunting license.

Trapping license, trap, transportation to and from trap site and bait

Same here a trapping license and if you have any luck fur tags so you can sell them or have a taxidermists mount them, they still need to be tagged first.