So, in addition to having a car, Tolkien also wrote about cars in two of his stories (one posthumously published, the other still languishing unpublished), as well as rather pointed remarks in one of his major essays.
The best known of these, of course, is MR. BLISS, which was probably (as Carpenter says) inspired by Tolkien's misadventures immediately upon buying a car of his own.* Here the car is the impetus for Mr. Bliss's adventures, the framework to get the story going, rather than the story actually being about the car itself.
Much more obscure is THE BOVADIUM FRAGMENTS, a short tale partly in English and partly in Latin. Here we have Clyde Kilby to blame, at least in part, for the tale's not being better known, in that when Tolkien was thinking against publishing it circa 1966 Kilby advised against it -- a pity, since an anti-car environmental message from Tolkien published in the mid-sixties might have done some good.
Almost (but not quite) as strong an anti-car statement can be found in OFS:
Not long ago . . . I heard a clerk of Oxenford [i.e., a fellow Oxford don**] declare that he 'welcomed' the proximity of mass-production robot factories,*** and the roar of self-obstructive mechanical traffic, because it brought his university into 'contact with real life'. He may have meant that the way men were living and working in the twnetieth century was increasing in barbarity at an alarming rate, and that the loud demonstration of this in the streets of Oxford might serve as a warning that it is not possible to preserve for long an oasis of sanity in a desert of unreason by mere fences, without actual offensive action (practical and intellectual). I fear he did not. In any case the expression 'real life' in this context seems to fall short of academic standards. The notion that motor-cars are more 'alive' than, say, centaurs or dragons is curious; that they are more 'real' than, say, horses is pathetically absurd . . ."
I think many Americans read this passage without realizing how specific it is: Tolkien is talking about the Morris motorworks out the Cowley Road, not far from where Tolkien eventually retired to on Sandfield Road; The Kilns also lays out that direction. Most of England's factories were in the north (Birmingham, Manchester, Sheffield), making the car-plants just east of Oxford the exception rather than the rule. It's ironic that the English equivalent of Henry Ford who set up his factories here should have shared the same name with one of Tolkien's heroes, William Morris (afterwards Lord Nuffield).
Two final snippets re. Tolkien and cars, while we're at it: if memory serves me rightly, Michael Tolkien in a radio interview recounts his anguish when in his youth they cut down all the trees (elms, I think) that lined Broad Street while expanding the road, presumably widening it to accommodate cars. Another reason for Tolkien to have no fondness for cars, especially given that he was living in a city perfect for foot-travel and bicycles. Finally, there's the further account in THE TOLKIEN FAMILY ALBUM that on family vacations during the years 1934-38, Edith wd travel by train with Christopher and Priscilla, John and Michael wd cycle down over the course of a few days, and JRRT "would drive Jo 2, weighed down with luggage, squeezing in himself at the last possible moment. He wd be surrounded by large numbers of Priscilla's soft toys, which she insisted shd share the holiday: on one trip someone asked him if he was a travelling salesman in teddy bears!" (FAMILY ALBUM, p. 64)
*this assumes we discount Joan Tolkien's claim that this story was modeled not on an actual car but on her husband's toy car from 1928. In any case, associational evidence within the tale proves that the story as we have it dates from circa 1932-36 (probably 1932/33), not to the late '20s.
**so far as I know, whichever of Tolkien's colleagues who said this has not been identified; Flieger-Anderson note that in the early drafts of OFS he is described as "the head of an Oxford college", which narrows the field somewhat.
***i.e., factories with automated assembly lines, not factories for building robots.