As indicated in prior posts, a complete history website for cars in general does not exist to my knowledge. Things like the Shelby Registry and the Boss 302 Registry have been around for some time and may have some information on those cars. Registries for other makes and models seem to be hit or miss with many "Registry" efforts fading away over time. Keeping in mind that 50 years have lapsed since these cars were built, time is generally not on your side. That being the case, here are several suggestions that might provide you with a tidbit of information that could open the door to the car's history.
For a run of the mill production car, your best bet is a methodical approach that includes an archaeological search. Starting with the Marti report (which provides production information from data purchased from Ford many years ago) that includes the original selling dealership. If they are still in business, I would start there to see if the original owner was a regular service customer. Another angle is to contact the DMV where the car is currently registered and tell them you are doing historical research on your vintage car (include a copy of your title as proof) and ask if they have any records connected to that VIN. At the very least request copies of registration and title documents in their files connected to that VIN. One could expect that the prior title surrendered when your title was issued should provide some useful information. Some states have request forms in which you demonstrate your legitimate reason for seeking the information before information will be provided. In some states, privacy laws prevent them from releasing personal data without consent of the person whose information is being sought. Offering consent for DMV to give your name and contact information to the prior owner(s) with a request to contact you, the current owner, might help legitimize your request. Note that in the past, if you knew someone in law enforcement, a friendly request might get you information. More recently, a number of jurisdictions have closed that loop citing privacy regulations.
Examine your car carefully.
Are there dealer licence frames?
Are there oil change stickers in the door jamb indicating where the car was serviced at one time?
Any inspection stickers or parking permits that connect the car to a particular place or time?
Any insurance stickers (AAA, State Farm, etc)?
Pull up the seat cushions to see if there is a random receipt, business card, or other item with a name/address on it. (I once found an old Driver's License under the seat of a car presumably belonging to a prior owner or at least a passenger!)
Check into the trunk drop offs to see if a prior owner left something down there with a name or address.
Check in the defroster vents as all sorts of things fall into the vent ducting.
Look behind the glove box for anything that might have spilled out (parking ticket/traffic ticket/receipt/envelope, etc)
Shine a bright light into the doors with the glass rolled down and look for anything that might be sitting at the bottom of the door (business card, credit card, etc)
Next, contact local Mustang clubs and attend local Mustang shows to see if anyone recognizes the car. A prior owner of the Mach 1 in my sig pic saw it at a local car show and exclaimed "That's my car" having recognized the license plate number. Long shot yes, but his recollection filled in some gaps in the known history (early 1970's) . A subsequent discovery of old DMV papers on the car confirmed his recollection.
Lastly, if you are committed to discovering your car's history, consider hiring a private investigator to dig around as they may have access to information that is generally not available to the general public.
*Principal wrench on this 69 Mach I*