From your article:
“The process to distribute the vaccine, particularly outside of nursing homes and hospitals out into the community as a whole, did not really exist when we came into the White House,”
Note the outside of nursing homes and hospitals, which were the targeted priority at the time.
"Biden plans to partner with state and local governments to establish vaccination spots in conference centers, stadiums and gymnasiums. The new administration will also deploy thousands of clinical staff from federal agencies, military medical personnel and pharmacy chains to increase vaccinations, and make teachers and grocery clerks eligible."
Even after he took office, it still wasn't available to the general public I guess, since they were, I don't know, prioritizing it or something.
“We’ve seen this factor all over the country where millions of doses have been distributed, but only about half have been given out,”
This also shows that the vaccine was available prior to Biden taking office, it was a shit show apparently, but still shows the tweet to be misinformation.
As I said to Jim below, the wording says that there was NO VACCINE AVAILABLE when he took office, and that is inaccurate.