Internalising an externality for a free merit good
Even if the vaccine is free it is possible to additionally internalize the externality with subsidy. That is government can just pay people to take the vaccine.  Note, that even with a flu vaccine just providing voucher for a free vaccine might not be enough to fully internalize the positive externality (or it could be too much this is always case dependent).
For covid-19 specifically, there is an interesting recent paper on optimal covid-19 vaccine subsidies (among other things) by Goodkin-Gold et. al. (2020).  Their model shows that optimal subsidy will depend on the reproduction rate of the disease. Higher reproduction rate makes stronger case for subsidies, while if the reproduction rate it sufficiently low there might be no need for the subsidies at all. Moreover, the authors show that optimal level of subsidies also depends on competition in the industry. Specifically, they show that generally the more competitive industry is the greater case there will be for per unit subsidy.