Always get a PPI (Pre Purchase Inspection)
This is good advice for any used car purchase, including a classic Corvette. Have a mechanic look over a purchase candidate and make sure they are familiar with Corvettes; since they are a common enough car finding a qualified inspector should not be difficult. There also are professional auto appraisers qualified to assess older Corvettes. If you are paying top dollar for a particularly rare or special Corvette, then doing a PPI is even more crucial. If it turns out that the car is, shall we say, "not quite" what it was represented as, then the value will drop dramatically. Let us put it another way. Even when an expert is considering the purchase of a classic Corvette, they will often have another expert look over the car and it's history to prevent them from missing something.
Check the documentation.
Ideally the seller will be able to offer complete documentation for their car. This would include original sales receipts, receipts for all maintenance and repairs and information on previous owners, etc. Photographs backing up the car's history should be available. Documentation of a car is very important and you'll see this reflected in selling prices. As with other aspects of classic Corvette buying and selling, the subject of documentation has some controversy. There are reports that fake documents have started to appear, complete with various techniques to "age" the material. Encouraging this practice is the fact that buyers put a premium on a good record trail.
It is possible that a good purchase candidate will only have minimal documentation; this can happen if previous owners were not aware of the value of good documentation or for any number of other reasons. The older the Corvette, the less likely that all the records were maintained and passed along. This is particularly true of Corvettes with many owners. But it is also possible that the lack of documentation is purposely done to hide problems.
Although it is reasonable that the early documentation of an older Corvette may have disappeared, its recent records should be available.
Do Your Research
Find out as much as possible about the type of Corvette you are interested in. Do more than just read the message boards and various Corvette email lists; post your questions to them. Find out the weaknesses and various "things to look for" regarding the Corvettes you are considering .
Join your local Corvette club
If you're not the club joining type, change your ways, sign up and be active. A Corvette club can be your best resource. You can learn a lot with first hand and in person contacts, including Corvettes available for purchase, recommendations for vendors and repair shops and so on.
Be Realistic with your Expectations
With the possible exception of Corvettes that have been recently restored by a qualified entity, almost all available classic Corvettes will, to some extent, have issues. If you won't consider anything with a tiny amount of dirt on it yet you are not willing to pay the top dollar that the best Corvettes can bring then your search may not come to a conclusion.