Do repair shops keep records of the cars they have inspected? If so would they have vehicle information such as VIN? And would they be able to search through these records to find a VIN if they are provided with vehicle make, model, year, and owner of vehicle?

Some shops wlll, others don't, many will have records under the customes name.

Also what inspection, do you mean state required inspections to get car tags?

Some shops wlll, others dont, many will have records under the customes name.

Also what inspection, do you mean state required inspections to get car tags ??

I am talking about the yearly required inspection in PA.

Like many businesses, repair shops are required to retain records for a period.
If they are required to keep the records so that they could be retrieved by way of "year make and model", they may have dozens of the same vehicle. While that could prove handy for your case, most don't keep records by VIN.
Your receipt would likely have the VIN recorded, but the repair shop isn't compelled to retrieve that number and provide it to you.
Why would they? If it's your car, you could easily get the number from the vehicle or your registration.
I'm not sure where you're going with this...