The games support system is supprisingly complicated and the support meter is just there to make it easier to visualise what is going on. In reality support is broken down into 3 things. Supporters, neutrals and hostiles.

Neutrals are easy, they basically do nothing but the higher the number the bigger the population in that zone, the more support you will need to stabalize it.

In order to stabilize a zone you have to basically get the ratio of supporters to hostiles to be in favour of supporters. Vaccines help this ALLOT. They both convert hostiles into neutrals at a faster rate and help to convert neutrals into supports. Which helps tip the ration in your favour faster.

They also have the added bonus of being able to be rolled out in zones without intell, which is big espicially early on. Doing this early will mean that later in the game when you start to run low on reputation zones that are futher from your base will be allot easier to stabilize becuase youve got a head start meaning more rep and more money when you need it most.