I made an offer for a vehicle on CarGurus and it was accepted. The seller has the vehicle on a lease and he will need to contribute money to satisfy the lease when I purchase it. How do we handle this transfer to make sure I am protected?

Don't go through with the transaction if in fact it is a lease. A lease is a rental agreement and the lessee does not own the vehicle. If the buyer has a loan and is paying on it still, then you can perform the transaction at the buyers loan office to make sure you get the title mailed to you.