I think it would work fine as long as the drug can be self administered.
The post office has all of the transportation infrastructure and distribution personnel to collect and distribute the medicine. Mail order drugs are already delivered by the post office every day to millions of Americans.
What may be necessary is for Congress to pass laws that would permit the post office to suspend delivery of regular mail if required to permit the exclusive routing of the drugs.
Other laws may be needed to use the latest census data to determine the number of doses to be delivered to each residence.