According to his biography, JRR Tolkien was indeed a driver, but not a good one.
In 1932, Tolkien purchased an automobile: a Morris Crowley. The car
was called "Jo" after the first two letters of its license plate, and
Tolkien had a tendency to drive it like a knight on his prancing
charger. On the family's first long-distance road trip, to visit
Hilary Tolkien on his Evesham fruit farm, the car's tires punctured
twice (a rather common hazard of the time) and Tolkien managed to
knock down a stone wall near Chipping Norton (for which he had no
one to blame but himself).
Tolkien's technique for navigating busy
intersections was to ignore all other vehicles and floor it, yelling
"Charge'em and they scatter!" as he blasted his way through traffic.
J.R.R. Tolkien: A Biography
He evidently sold his car in the 1940s, stating that they were turning his beloved Oxford into something resembling Mordor
"Though, the spirit of ‘Isengard’, if not of Mordor, is of course
always cropping up. The present design of destroying Oxford in order
to accommodate motor-cars is a case."
“It is full Maytime by the trees and grass now. But the heavens are
full of roar and riot. You cannot even hold a shouting conversation in
the garden now, save about 1 a.m. and 7 p.m. – unless the day is too
foul to be out. How I wish the 'infernal combustion' engine had never
been invented. Or (more difficult still since humanity and engineers
in special are both nitwitted and malicious as a rule) that it could
have been put to rational uses — if any.”
Letter #64 to Christopher Tolkien
As an aside, it's worth noting that since he began driving before 1932, he wouldn't have held a driving licence since they were only introduced in 1934 and stopped driving before they became mandatory for all drivers.