The colloquial noun vee-dub (also Veedub, Vee Dub, etc.) designates a Volkswagen car.

In this noun (from the pronunciation of VW, initialism from the name Volkswagen), dub is a shortened form of the adjective double in double U.

Note: Likewise, the verb dub (meaning: to provide an alternative soundtrack to (a film, etc.)) is a shortened form of the verb double.

The noun vee-dub has probably, in the course of time, been coined on separate occasions by various persons, independently from one another. The earliest occurrences that I have found are as follows, in chronological order:

1-: From the column Teen On Wheels, by Michael Lamm, published in The Sheboygan Press (Sheboygan, Wisconsin, USA) of Monday 18^th December 1967 [Vol. 61, No. 1, page 20, column 2]:

DEAR MIKE: I heard Ralph Nader, the safety expert, talk here at school the other night. He said that one of the unsafest cars on the road is the Volkswagen. I own a VW beetle, and I’ve never really felt any less safe in it than in any other car. Is the Veedub really so unsafe?
—HMMMM

2-: From an advertisement for John Palmer Motors, published in The Canberra Times (Canberra, Australian Capital Territory, Australia) of Saturday 9^th August 1969 [Vol. 43, No. 12,386, page 8, column 1]:

1968 VW 1600 SEDAN
Step into this late model vee-dub at a considerable saving on new car price ….. $1790

3-: From The Front Room Boys : A Play in Twelve Scenes, by the Australian playwright Alexander Buzo (1944-2006), first produced by the Australian Theatre Co. at the Australian Arts Laboratory, Sydney, on Thursday 9^th October 1969, and first published in Four Australian Plays (Ringwood (Victoria): Penguin Books, 1970) [pages 48-49]:

GIBBO: Get out of it. They’re gutless, those cars. They’ve got a six-cylinder body on a four-cylinder engine, and a hydraulic camshaft that buckles under the carbie, haven’t they, Presto?
ROBBO: Pigs.
GIBBO: They’ve got all the defects of the Vee Dub fifteen hundred and none of its virtues. And they slip their clutch to boot.

4-: From Doug’s Motor Notes, published in The Noosa District News (Tewantin, Queensland, Australia) of Friday 20^th March 1970 [Vol. 2, No. 15, page 6, column 1]:

‘Herbie’ the Love Bug has come to Noosa and is appearing at the Drive – In this weekend. We saw the trailer and its [sic] a beaut show for kids of all size’s [sic]. If you are a Vee-Dub driver its [sic] a must. I pity all other owners after you see Herbie.

5 & 6-: From the Springfield Daily News (Springfield, Missouri, USA):

5-: Of Friday 29^th January 1971 [Vol. 81, No. 25, page 30, column 8]:

Volkswagen Group To Meet Sunday

The Vee-Dub Club of Springfield, composed of Volkswagen and dune buggy enthusiasts, will hold its weekly meeting Sunday at 3 p.m. at Sho-Mee dune buggy center, 3240 West Sunshine.
The club is seeking more members. Interested persons are welcome.

6-: Of Wednesday 10^th February 1971 [Vol. 81, No. 35, page 14, column 3]:

Vee-Dub Club Looks To Join With Joplin

The Vee-Dub Club of Springfield, composed of Volkswagen owners, plans a meeting at 2 p.m. Sunday at the Sho-Me [sic] Buggy Center on West Sunshine.
Club members then will go to Joplin to investigate forming a Southwest Missouri Association of the organization. Various activities, including rallyes [sic], are planned later in the year.

7-: From a review of the Supervolks Zinger, “a full size version of MPC’s wild new Zinger”, published in The Sioux City Journal (Sioux City, Iowa, USA) of Sunday 9^th January 1972 [page B8, column 4]—MPC stands for Model Products Corporation:

The completely hand built body is exactly one-half the size of a regular “Vee-Dub” and powered by a monstrous dual-blown late model Ford engine. A hefty Hilborn injector system poking through the roof tells folks there’s more to this bug than first meets the eye.

8-: From a review of the Volkswagen K70, by Dave Baxter, published in the Evening Post (Bristol, England) of Thursday 16^th March 1972 [No. 12,313, page 38, column 7]:

It gets a bit noisy inside but this should make Veedub fans feel at home.