Na. Lots of places have socialized medicine like Israel. Not many of them, if any of them, are having the successful rollout Israel is enjoying. The entire Israeli society is prepared for this sort of event more so than any other in the world. Civil society, healthcare (as you pointed out), military and, dare I say, even religious all work together to pull this off.
I agree with Nir, HMOs are carrying a huge amount of the load here (after the government supplying all those vaccines). You have 4 HMOs total -- one of them (Clalit) being huge -- so there's not much fragmentation; a very fast crew of health workers which allows us to reach that level of vaccinations/day; and an existing well-tested infrastructure for flu vaccines, which are in high demand each year.
I don't think the military was involved yet, and religion was in fact working against vaccination efforts as a few anti-vax rabbis drove sectors of the Haredi population off.
EDIT: Saw another comment about a person in uniform doing the immunization so I might've been wrong about the army's involvement. But either way most comments referring to military involvement mean i.e. wartime levels of involvement, military handing out gas masks door-to-door etc. We're not at the level that the army is a sizable force, from what I can tell. (Except immunizing their own soldiers of course)
