We eat mainly fresh (one of the benefits of living in Iowa is a plethora of fresh produce not only in the summer, but also in the winters due to the large number of greenhouse farms) but my son absolutely adores frozen peas (he eats them by the bowl-full, still frozen, like they're ice cream) so we buy a bag or so of those each week from Trader Joe's. We'll also occasionally buy frozen haricot vert if we can't find good looking fresh ones.
The most disgusting vegetable I ever ate was a frozen Brussels sprout. It was absolute mush even after roasting. Blech!
The only canned "vegetable" we buy is pozole for a delicious pork and pozole stew that my wife makes, but that is because it's the only way we can find it locally.
ETA, I lied, we also use canned tomato sauce. Not sure if that counts, but don't want to give the impression we make our own tomato sauce.