To return to the OP: within the limits of my knowledge, there are basically two kinds of "part time volunteers" in a military context.
The first is a Reserve system. This usually consists of a mix of retired ex-military who are keeping their hand in (often joining the Reserve is required when you retire) but have mostly moved on to civilian work, and people who join for the "adventure" or the paycheque or out of national pride, but don't want to commit to full-time military service (or can't qualify). Reserves are equipped by the state (meaning country, not an Australian or American political subunit) and primarily exist to provide a pool of spare bodies that could be called up for active service in an emergency, as well as to create a pool of skilled instructors and trainers (because Reserves tend to be high turnover) to make any need for massive recruitment much faster.
It's fairly possible that something like this could exist with Star Fleet. There's plenty of evidence that it's hard to get into Star Fleet. A local, planet- or system- wide organisation with sublight ships and other equipment provided by the Federation and a level of training adequette to deal with a narrower range of tasks could be very useful for both military defence and disaster/humanitarian response.
Whether or not the Federation is supporting such a body (and post-scarcity for the individual is much different from post-scarcity for an interstellar polity) feels very time-frame dependent. During TOS, with internal tensions high and conflict with the Klingons never far away, planetary reserves seem likely. Following the Dominion War, likewise. During First Season TNG when the Federation hasn't had a major conflict in ages? Far less likely that there's a Reserve system in place and if it is, it's probably undersupported by both government and the public.
The other kind of "part time volunteer" is the Militia. A Militia is a set of volunteers that provide their own equipment and are mostly responsible for their own training, though the state may provide some resources if the group maintains some certain minimum standards. The US National Guard is not a Militia, it's a Reserve, and off the top of my head I'm not aware of any G20 country that still uses militia, because they're unpredictable and their training and procedures are unreliable. Generally speaking, militia were used when soldiers weren't highly skilled professionals themselves, and when local groups felt a need to band together to protect themselves, governments found it easier to control this by giving them an official nod.
I don't think there would be any official militia in the Federation. There may be unofficial ones - one could argue the Maquis are an unofficial militia - but I can't see a group of civilians, being mostly left to their own devices, getting any sort of recognition from the Federation.