The White House is shipping nothing. Any more than it shipped Pfizer and Moderna vaccines in December. However, what Pfizer and Moderna have shipped and will continue for months to come was due to work and purchases done by Trump and his OWS team. The same is true of Johnson & Johnson. Their success was aided by Trump and his OWS team, and the first 100M doses they ship - capable of fully vaccinating 100 million people, not "just" 50 million - are under a purchase contract negotiated by Trump and his OWS team.
To date, the only vaccine thing Biden has done is to exercise additional 100M dose options with Pfizer and Moderna ... under the contracts negotiated by Trump and his OWS team. Not insignificant, but nothing spectacular.
The J & J vaccine has two unique characteristics compared to Pfizer's and Moderna's: it is single dose, one-and-done; it can be stored at ordinary refrigerator temperatures.
If Biden & Crew wanted to accomplish something only begun under Trump, they'd push for all the vaccines to be tested with youths, children, and pregnant women. Moderna started a Phase 2/3 test with youths 12-17 in December, working with BARDA, a division of HHS. Pfizer began a Phase 2/3 test with pregnant women (after first completing a developmental and reproductive toxicity (DART) study) February 18th, but the only US government involvement I've found is writing the test protocol.