yes, states are largely in charge of planning their own vaccine distribution efforts. Even as Biden's team will try to federalize more of it, the bulk of the effort will still rely on the states themselves:
"The Biden campaign and transition team have planned for a more prominent role for the federal government in the U.S. COVID-19 response, which would likely include more detailed federal guidance and a stronger federal hand in vaccine distribution, planning and implementation, even as state and local jurisdictions will remain responsible for much of this effort."
https://www.kff.org/coronavirus-covid-19/issue-brief/states-are-getting-ready-to-distribute-covid-19-vaccines-what-do-their-plans-tell-us-so-far/