Yes, a mobile home constructed prior to June 15, 1976 will have a serial number. It could have been etched into the front metal frame or in the tongue. If it had been etched into the tongue and the tongue has been removed--the physical evidence is gone. However, when the home was brand new and sold by the manufacturer to a dealer or a home owner, a Manufacturer's Statement or Origin/Certificate of Title would have been issued. On that MSO/title would be the serial number. Any time after that when ownership of the home was changed the serial number would be referred to again. If Certificate of Title was surrendered or purged when taxing of the home became real property, that serial number should have been in the paper work somewhere. On this site is a link for the procedures in each state of how a mobile/manufactured home is changed in assessment records from personal property to real property. There is a record somewhere. Check out the procedures in the link for your state, then start digging through recorded documents, assessor, tax collector, building, zoning, etc, etc offices. Some government agency has that serial number somewhere in their records.
Mobile homes constructed prior to June 15, 1976 will not have HUD labels/tags/decals/insignias. They might have a decal or tag from an inspection agency for that area or manufacturer at that time. There even might have a data plate or similar piece of paper that would report who, where and when it was built. Typically that piece of paper was posted inside of a cupboard, cabinet, closet, or on the wall by the electrical panel. It might have a map of the USA on it. But those were not required prior to June 15, 1976, and not every manufacturer was inclined to post them.
By the way, although the mobile home (if constructed prior to June 15, 1976) or manufactured home (constructed after June 15, 1976) has been remodeled, it is still considered a mobile/manufactured home by almost all lenders. Remodeling, renovation, reconstruction, etc does not change the original building code that was followed in the factory.