Hi, Dave. The first thing to do is to take your chair to a reputable antique dealer and get an appraisal. Since you don't know the history of your chair, and since you can lose thousands of dollars in refinishing some antique furniture, it is a wise idea to be sure.
If your chair is in the "nice but nothing special" category, you can refinish. Today, it is easier to strip and refinish than to reamalgamate the finish, in my opinion. Start with a relatively safe stripper like Citristrip and remove the old finish. You will probably want to use a stripping pad, sold by 3M, Citristrip and others, for most of the work but you may want to use a putty knife on the seat and back. Be sure to follow the instructions on the label and give the stripper plenty of time to work. Also, be sure to wear chemical resistant gloves and eye protection. When you have stripped, you can refinish with the product of your choice.
Good luck and let us know if you have further questions. Henry in MI