Searching for parts for a collector's car is going to be a long, arduous process no matter how you go about it. Even if you get into contact with a specialist at finding those parts it could be months, if not years, before they get what you need. The reason for this is not laziness but just availability. Due to the ever-decreasing supply of antique car parts, they are increasingly difficult to acquire. Your best bets to getting refurbished parts on your own is to explore barn sales and other auctions that may contain old cars, to explore hobby message boards dedicated to collector's cars, and to visit junkyards. Even with a lot of time and dedication you may find yourself searching for months before you locate what you need.
For more modern and available parts, the primary reason to get a refurbished piece is to save money. Compared to buying new parts, refurbished car parts can be a fraction of the cost and operate a similar quality level. To find refurbished parts for your car you go to your auto dealer and see if they stock refurbished pieces. If the prices are too high or they do not then your next best bet is to look for a business that specializes at refurbishing car parts. You may not be able to find one locally so check online. A quick search for refurbished parts for your car model should turn up plenty of results with lots of businesses more than happy to sell you the part. Be sure to check how reputable a seller is before making a purchase though. If the part turns out bad after a few weeks then you are often left with very little recourse. Make sure that an online seller has a large history of sales and few to no complaints about a part dying shortly after arrival.