Wikipedia gives me the impression that Appalachian English is a member of the Southern U.S. English dialect collection and can be subdivided into a southern variety called Smoky Mountain English and a northern variety called Western Pennsylvania English. The construct you ask about appears common in both those regions.
Dictionary of Smoky Mountain English by Michael Montgomery contains a brief summary of grammar and syntax of Appalachian English which contains a sample sentence:
That thing needs washed.
In contrast, some other sources suggest that the prevalence of this construction is rather characteristic of the northern variety of Appalachian English, and that it is also accepted as intelligible/correct in a much wider territory, especially in Midwestern American English. You can further gauge how deep you are into this usage; some speakers use this construction only with "need", others with "need" and "want", and the most devoted users say it like this with "need", "want", and "like".
Elspeth Edelstein, in her article "This syntax needs studied" (where she characterizes the variety as Pittsburghese, a synonym for Western Pennsylvania English), suggests that "[this construction] does not arise from phonological ellipsis of to be, as is often assumed." However, this is a synchronic study, not a diachronic one, so it is more concerned with how the construction works in then-current language than with what it developed from. In her assessment, the participle ("washed") is verbal, rather than adjectival; i.e., it refers to the deliberate act of washing rather than to the ultimate state of the car.
Christopher Strelluf, in his article "needs+PAST PARTICIPLE in regional Englishes on Twitter" documents the usage of the construction (apart from above discussed regions within the U.S.) in Scotland, Belfast, Newcastle. Some of the previously cited sources also document that the construction is much more prevalent among whites than among blacks within the U.S. Taken together, it appears quite possible that the construction originated in northern parts of British Isles and arrived to the U.S. through the Puritan wave of immigration.
Of only anecdotal value, because Celtic modal constructions are very diverse: the simplest equivalent of "The car needs washed" in present day Scots Gaelic would be "Feumar an càr a nighe." ("Need-the-car-to-wash".) See, there's no equivalent of "to be". (Some resources actually translate feumar as "needs to be" rather than just as just "needs", but that's mainly because they are translating into English.)
It is quite conceivable that the construction initiated from language contact with Gaelic and Irish and spread to your birthplace through immigration either during the 17th century and/or later. If we assumed that the syntax didn't exist in the U.S. by mid-19th century, we might have some difficulty explaining its wide acceptability in present day Kansas or Montana, per research done by Josephine Holubkov. German (or German Jewish) immigration in the second half of the 19th century doesn't seem to fully explain this geographic spread of the syntax within the U.S. either, let alone its spread within British Isles). So it's probably older than that.
While I chose to look at Scots Gaelic, there is no reason to assume that the syntax was borrowed exclusively from contact with a single language; and that that language was Scots Gaelic in particular, and that the single modal verb I looked at played a particular role. However, I'm convinced that the wide availability of the syntax within parts of British Isles is the smoking gun and that Celtic languages are probably implicated.