Limited supply does not mean unavailable, it means limited availability. It was approved, and it was being distributed before Biden took office. They were prioritized, as they should have been for the initial rollout. If you want to argue semantics and say that the general public could not get the vaccine because of this, sure, have at it. But I find it rather sad for a White House who is headed by someone who got the vaccine a month before he took office, to claim that it was unavailable.
Again, the statement says that there was no vaccine available. It was available for medical personnel, seniors, and other high risk people before he took office. It's inaccurate.