Usually, when some used car lot operations "dyed" seats and/or carpets, it was to renew the colors, not change them completely. Some of the OEMs sold dyes to match their factory colors, for such purposes. How good it looked could result in how well the operator applied them. It would be easier to dye the carpets, though, with less probability of the dyes later transferring to the occupants' clothes on a hot and humid day. YIKES!
The bulk of Chrysler trim codes, until about 1973, had a metallic or pearl tint to them. Something one might miss (I did, although I knew they looked better than normal) until a similar color from a non-Chrysler Corp vehicle is placed onto the vinyl or fabric for a color-match comparison. THEN it becomes very apparent. As this coating ages, as on white, it can appear "dirty" and will not clean up with cleaners, from my experiences.
I came to realize why the Chrysler trim codes looked better when a trim shop owner in Lubbock, TX showed it to me, back in about 1973. The difference was significant on the lighter-brown vinyl of the '66 Newport Town Sedan. I do not know of any dyes which will duplicate that effect.
SEM and possibly others make some "vinyl spray coatings" for seats. From what a friend told me back in the 1980s, those coatings tend to be like sheet metal paint, cleaning prep is important. Even with great application, the additional coating can make the vinyl feel stiffer and have a non-vinyl sheen. Hopefully more recent formulations can lessen these things?
Even if you could find some Rit Fabric Dye somewhere, it would be a messy operation. You, the seats, and surroundings could end up being different colors. Might be fun and amusing, but still "a mess".
Just some thoughts,
CBODY67