I don't know how it happened but I have broken one of the wheel studs on my right wheel!! Can I fix this myself or would it be better to take it to a shop??

Unless you have some specific tools and experience on the Corvette wheel studs I would take it to the dealer. I messed up a few of them on my 01 ZO6 and I thought it would be no problem to replace them (like the old days) but they weren't so easy. If I remember correctly I had to borrow a huge socket to pull the wheel hub, but it has been a few years so I dont remember for sure.

If I remember, my buddy had this problem and he tried to do it himself and when he got the tire off he found out that there was no acces hole to put the new stud through. So he had to take off the entire brake and remove that shield thing. Now I know he was not an expert but he has always had a nack with cars and he said he didn't know why gm didn't make an acces hole?