It might be worth double checking exactly what it is they want.
If they're submitting the entire system for UL approval, it may well be that they just need the bare PCB itself to be a UL recognised component. This is perfectly normal, and most PCB manufacturers will already have the necessary approvals. You just need to put a note on your PCB drill drawing that shows the position for the mark, and an instruction for the manufacturer to add the UL Recognised Component ("UR") mark, along with the 94V-0 flammability rating, and their own logo or other mark to indicate the bare board supplier.
There's usually no additional cost to you for doing this. You just need to tell your PCB supplier that you require the marking and traceability, and if they can't do it, then switch to another PCB supplier who can.
If they want your whole PCB assembly to be a UL recognised component, then that's a can of worms. It will be expensive, and intrusive in terms of ongoing inspections and documentation. Saying 'no' might well be the right answer. Increasing your price substantially to cover your costs might be another good option.
In practical terms, UL are most concerned about fire. They'll insist on testing, or seeing test reports for, any parts of your board assembly which look like they could start a fire, or support combustion in the event that one breaks out elsewhere. Individual small electronic components are't really a concern, but you'll need to ensure any plastic coated wire or connectors are UL recognised, and that any power supplies have inherent limits on power output such that they can't start a fire in the event of a fault.