Inventing a vaccine is the first step, getting billions of does out acrsoss the globe is the following task.
This is the plan in the US, via Politico:
Nationwide distribution will be a "joint venture" between the Centers for Disease Control and Prevention and the Department of Defense
Citing a senior administration official.
Department of Defense "is handling all the logistics of getting the vaccines to the right place, at the right time, in the right condition," the official said in a call with reporters, adding that CDC will remain in charge of tracking any side effects that emerge post-vaccination and "some of the communications through the state relationships [and] the state public health organizations."