We reside in West Virginia and have recently bought an auto. Do new cars require a tech inspection or not?
Of course. A car registered in West Virginia should carry an inspection sticker to comply with state regulations. And in case the car vendor did not provide you with such a sticker when you bought your car, then it's your responsibility to go through the inspection and get the sticker on the window.
The WV Division of Motor Vehicles imposes any car whether new or old to be inspected within three days in case it doesn't carry the respectful sticker.
West Virginia Code 17C-16-9 provides data on the requirements to be met during the inspection, as well as penalties for not meeting them. Here's an exempt:
It is considered as an infringement for the driver and/or the owner of the vehicle to fail in representing a valid proof of undergoing technical inspection, which is prosecuted respectively. Car sellers are not obliged to provide such proof in case the new vehicles are driven on operational or demonstration purposes only and therefore it is the responsibility of the purchaser to obtain such a certificate.
Not carrying such a current and valid certificate imposes a fine of not more than one hundred dollars and requires the car owner to present a receipt of undergoing the required inspection within three days after receiving the fine.
In West Virginia such certificates are issued for a period of twelve months and can be obtained after undergoing inspection at any official inspection station licensed by the West Virginia State Police. The sticker you will receive after the inspection should be placed on your windshield otherwise your vehicle is illegal to drive within the state.
New West Virginia residents have 10 days at their disposal to run through the inspection after entitling the vehicle within the state. This also applies to autos bought outside the state. Having an inspection ticket from a nearby state (Louisiana, Mississippi, Missouri, New Hampshire, New York, Oklahoma, Texas, Utah, and Wyoming) is considered to be valid too.
It always helps asking the vendor about the inspection sticker when buying a new car. The dealer should tell you whether they already have obtained such a ticket for the vehicle you buy or you have to get it on your own.

