We really have two different areas of concern here-the suckers and the chance of oak wilt. We have had many questions about the root suckers of various trees, including oak, recently and will select a couple of those questions that apply best to where you are. We can at least relieve your mind about the tree getting oxygen from the suckers. If a tree has been injured and is dying, often it will put up suckers to provide some leaves to manufacture food for the still-alive roots beneath the ground. If you have a healthy oak, it is much better to just cut off the sucker. However, you must remember the cardinal rule about protecting oaks from Oak Wilt-don't prune anything from February to June, as that is when the insect vector for oak wilk, the nitulidid beetle, is active. So, get after those suckers now, and quit in mid-January. The first previous question we want to refer you to is discussing oak wilt, and we encourage you to follow all the links and find out all the problems you might face and where to get help.
From another previous answer, also from Austin, here are our recommendations on getting rid of the suckers:
You are probably doing the best thing in just clipping those little suckers off at the ground. They're unattractive and as time goes by, you could have a whole grove of teeny tiny live oak trees, which is probably not what you had in mind. And that could be a problem, too. In a wild, untended situation, this clonal method of reproduction could result in a grove of trees commonly referred to as a motte, with interlocked roots. Oak wilt, one of the biggest attackers of the live oak, can be spread through that root system. Prevent developing an "oak wilt center" by getting rid of those little sprouts.