You can see it as a metaphor: when you are deprived of food, your stomach is supposedly empty, and then you will tighten your belt to make sure your pants will still fit. This is what the allusion to a ceinture means: to have to do without some resources.

The standard phrase is se serrer la ceinture, but you can also use faire ceinture or just ceinture, as an interjection.

Les budgets ont été réduits, il va falloir se serrer la ceinture.

J’ai plus de sous pour les vacances, on va devoir faire ceinture.

je t’ai dit que je n’ai plus de sous ! Ceinture !

So in your example, it simply means that if the neighborhood does not vote for the mayor, they should be prepared to have less of what was promised, or even none of it (money probably ?).

On a humorous tone, note that ceinture can also mean you are deprived of … sex:

Je me suis engueulé avec ma fiancée, alors ce soir pour moi c’est clair: ceinture ! (= no sex for me !)

See the entry in Wiktionary, in particular for the interjection.