All of the variants, including the original Wuhan strain, can infect vaccinated people. The vaccines have never been 100% correct. However, across the variants, Delta included, breakthrough infections are, much more often than not, akin to mild colds. Another tidbit is that, at least with the OG strain and the Alpha variant, breakthrough infections also carried a much smaller viral load, meaning that the person is much less infectious. Could be the case with Delta as well, could not. We don't know yet. Boosters could also possibly help, if it is the latter. But again, we don't know. We're still figuring this out.
But I think that one thing most reasonable people know is that continuously locking down, and closing businesses (or, in our industry's case, cancelling weeks worth of performances), is not the answer. This virus is going to become endemic, so we're going to have to learn how to live with it. The more population immunity there is, the less noticeable breakthrough infections are.