As is the case with state labor federations, there is no rule saying that a local must belong. An individual international union might have a rule that its locals must belong to central labor bodies, but the council itself can't make such a rule. They have no authority over the local unions in their areas. Most officers of central labor councils are volunteers. Outside of large cities, few councils have paid organizing staff.
Some councils have a building, others operate out of their presidents' homes.
A central labor council president can often give you information about unions in the community. Political Interests Labor organizations are highly political in two senses of the word. Citizen organizers should know the history here. Usually, elected officials who are strong supporters of labor are also good on other citizen issues, and vice versa, but this is not necessarily always the case. If your group has opposed or pressured a politician supported by local unions, it is very important to be aware of this when asking them to help you.