This is one of those technically-true-if-you-ignore-the-context claims.
Yes, the Institute Dr. Fauci leads started working with Moderna as soon as the SARS-CoV-2 virus (as it later came to be called) DNA sequence was leaked from China. However, the Trump Administration had already offered help to China for their effort against the outbreak (not knowing, at the time, that China had delayed telling the world what they knew and was deceiving the world about some of what they knew). However, the NIH, in 2020, including the Institute Dr. Fauci lead, was part of the Trump Administration, not a separate entity.
Yes, the Institute Dr. Fauci leads started working with Moderna without input from OWS, but that ignores the fact that OWS didn't yet exist.
Yes, the Institute Dr. Fauci leads started working with Moderna without input from OWS, but that ignores the fact that within weeks of the DNA sequence becoming known, Trump organized the various efforts of Federal agencies - including the Institute Dr. Fauci leads - into one coordinated taskforce.
Yes, the Institute Dr. Fauci leads started working with Moderna without input from OWS, but that ignores the fact that but for the taskforce and OWS funding and accelerating the vaccine testing and approval processes, none of the vaccines currently approved would yet be, and would not be for many months.
Yes, the Institute Dr. Fauci leads started working with Moderna without input from OWS, but that ignores the fact that but for efforts by the taskforce and OWS in which none of the NIH were involved, the various necessities for the vaccination program - vials, syringes, needles, disposables - would be in short supply, a program-stopping bottleneck.
Yes, the Institute Dr. Fauci leads started working with Moderna without input from OWS, but that ignores the fact that through the taskforce and OWS, many Federal agencies worked with multiple vaccine developers in various way. Dr. Fauci's Institute had zero to do with Pfizer, the first vaccine to be approved in the US. HHS' BARDA, not any of the NIH, started the work with J&J/Janssen Research.
Dr. Fauci played a real part in the Feds' vaccine effort, but his contribution was neither the whole effort nor even a majority of it. He's vastly exaggerating his contribution to the effort, basically claiming it was HIS (not The Big Bad Bogey-Trump). The MSM, even the Daily Mail apparently, are too damned ignorant to spot and call out Fauci's Lie-By-Omission.