Could those be vinyl colors pulled from a vinyl color palette on Flexi or Signlab? If the design is simple it might have been from the days of cut vinyl, or it could be a palette from the day's offset inks?
Knowing what it is may not be of any help if there isn't a viable way to translate it into CMYK or PMS. So it might be easier to call the customer and ask for several samples for something to match (business cards, letter head, brochure, vehicle graphics...) Chances are if you get a few different samples you have an opportunity to educate them about CMYK vs other technologies by showing them the color differences in their own existing branding. This will help you avoid the color matching rabbit hole and keep you from spending several hours trying to get an exact color match on $200 order. It's a losing proposition for all involved.