Arguments like this are, unfortunately, far too common. When someone doesn't like what another person or group is doing, he or she often claims that the behavior is "unconstitutional." First, the U.S. Constitution applies only to government activity (with the exception of the 13th Amendment's prohibition of slavery), so the efforts of a group of private individuals cannot be unconstitutional.
Second, the recall is proceeding according to state statutes setting out the procedures for a recall election. Whenever there is a procedure such as this, it generally constitutes what is called "due process of law." (In this case, it would be specified as "procedural due process.") Such procedures are constitutional unless they have absolutely no reasonable basis. Here, Arizona's recall statutes are fair and evenly applied. There is no reason to claim that they are unconstitutional.
Those are the two primary reasons that Sheriff Joe's supporters are wrong. There are other reasons (including the fact that Sheriff Joe has no constitutional right to remain in office without facing a recall, and the fact that the citizens' right to hold a recall election is set forth in the Arizona Constitution, among others), but these suffice. Is the recall petition drive something that harasses him? Perhaps. But it cannot be considered unconstitutional.