Yes. There are two types of elections associated with unions. There is an initial election to determine if the majority of employees in a bargaining unit want to be represented (or discontinue representation) by a union or whether or not to switch their union affiliation.
Once the initial election establishes the union's presence in a particular workplace, then the union is supposed to be run democratically, where majority rules - so there will be elections and voting to choose officers and make decisions. For more information about democratic principles and unions, see the Association for Union Democracy.