On my 1960 Morgan 4/4 Series I, the chassis number is located on the driver's (RHD) side of the chassis cross-member at the rear of the cockpit adjacent to the transmission tunnel cover.
Yours may be to the left of the tunnel cover. It should be in the range A590 to A648 if a Series III; starting B650 for a Series IV.
The other posts regarding the body/engine number. They should be stamped on the bonnet flanges for a car of your era. Apparently this same number was stamped on the bodyframe although I have yet to find it on my Series II.
By the way, VIN plates were never fitted to Morgans until later. My 1976 Plus 8 has one.
Finally, what engine is fitted to your car? Is it a Series III or Series IV?
Last edited by Cheshire_David; 29/11/17 09:18 AM.