Barry,
Here are the applicable Calif. Vehicle codes. Your plastic windshield will pass DMV inspection however, no wipers will cause some grief.
In most instances, the vehicle will have to pass inspection and I serious doubt that they will allow the vehicle without wipers.
26700. (a) Except as provided in subdivision (b), a passenger vehicle, other than a motorcycle, and every bus, motortruck or truck tractor, and every firetruck, fire engine or other fire apparatus, whether publicly or privately owned, shall be equipped with an adequate windshield.
(b) Subdivision (a) does not apply to any vehicle issued identification plates pursuant to Section 5004 which was not required to be equipped with a windshield at the time it was first sold or registered under the laws of this state, another state, or foreign jurisdiction
26706. (a) Every motor vehicle, except motorcycles, equipped with a windshield shall also be equipped with a self-operating windshield wiper.
(b) Every new motor vehicle first registered after December 31, 1949, except motorcycles, shall be equipped with two such windshield wipers, one mounted on the right half and one on the left half of the windshield, except that any motor vehicle may be equipped with a single wiper so long as it meets the wiped area requirements in Federal Motor Vehicle Safety Standards Governing Windshield Wiping and Washing Systems.
(c) This section does not apply to snow removal equipment equipped with adequate manually operated windshield wipers.
26707. Windshield wipers required by this code shall be maintained in good operating condition and shall provide clear vision through the windshield for the driver. Wipers shall be operated under conditions of fog, snow, or rain and shall be capable of effectively clearing the windshield under all ordinary storm or load conditions while the vehicle is in operation.