Could, for instance, Florida block Florida State from entering the SEC? Not on its own, it can't.
All of the noise about Texas A&M joining the SEC has naturally brought up speculation about who would join with the Aggies. A 13-team league is unwieldy, and it makes normal scheduling an impossible task.
All of the people putting out the Texas A&M rumors agree that the conference would look to add another team to the East division rather than pick up two Big 12 teams and move someone from the West to the East. That is backed up by reports from Chuck Carlton of the Dallas Morning News who is saying that no one has plans to follow A&M and that the Big 12 will soldier on with nine in the short run.
Some of the candidates being thrown around have rivals in the conference already. That fact leads to a lot of talk about how Florida would block FSU's entry or Georgia would block Georgia Tech's entry. Unfortunately for those theories, individual schools cannot deny others' entry on their own.
Page 3 of the SEC Constitution covers membership. One of the bylaws on the matter states:
3.1.2 Granting of Membership. Membership may be granted by invitation of the Conference at a regular or called meeting. A three-fourths vote of members is required to extend an invitation for membership. [Revised 5/30/91]
Only nine teams need to approve of a school in order for it to join the league. One school cannot block someone's entry, but four can.
Coincidentally, four SEC East teams have annual non-conference rivalry games: Florida, Georgia, Kentucky, and South Carolina. If they all were adamant about not wanting their rivals to join the SEC, they could form a voting bloc that could successfully prevent entry for FSU, Georgia Tech, Louisville, and Clemson. I have no idea whether they would consider such a pact, especially given that GT and Louisville aren't likely to be serious candidates, but that's one scenario where the schools could keep their in-state counterparts out.
The point is though, one school cannot block another school from joining the league by itself. They need to find three friends of the same mind in the conference in order to get it to happen.