Hi Cardinal2B

Yes 1948 Chevrolets DID have a VIN/serial number.

I'm not interested in getting into a discussion over the semantics of what is or isn't a "VIN" as opposed to a serial number.
Suffice it to say, Chevrolet was calling their "vehicle identification numbers" exactly that well before any government legislated numbering system came into force.
Either way, your car has/had one when new, and still will unless it was removed by someone.

I'm not sure exactly where yours is/was, but I'd look on the door post/jamb at the front edge of the driver's door.

For more information on VIN/serial numbers in 1948, look here:
https://chevy.oldcarmanualproject.com/chevyresto/4813.htm

Hope that helps. 🙂

PS
While it is true that many states used the engine number to title cars, many did not.
When they didn't, they used the serial number/VIN.