You can readily wire money with any bank in the US. It exists but people don’t really use it. It just never caught on much as a consumer product because of alternatives like ACH and debit cards that achieve the same thing.
That's mostly because it's slow and most banks charge a fee ($35 or so). The only time I've used it is wiring money to the escrow company when buying a house.
Regular car dealers just aren't set up to receive wire transfers. They wouldn't know what to do with it.
Their banks can receive wire transfers just like any other bank.
It's not a bank issue. The retail F&I staff at most mainstream dealerships aren't accustomed to dealing with wire transfers and literally don't know what buttons to push. You can probably get it done eventually but it will involve some delays and complaints.
Ask them about chip and pin. Unless things changed during covid, I recall chip and signature.