It depends on the state. Some states, like Florida, have "right to work" laws that can more accurately be described as “right to have workers” for employers because in these states workers do not have the right to strike. As a result, unions do not have as much power.
There can be significant differences in the law between the NLRA and state law regarding public employee unions, but state laws vary considerably, and we do not know what state you are referring to. In Pennsylvania, public employee unions have been permitted to organize since 1947, for example, as a matter of statute. If 30% or more seek recognition as a union, an election is requested, and the Pennsylvania Labor Relations Board holds a hearing to determine whether or not an election will be held. A majority of the votes in an election leads to the next step, which is a certification of the union.