The group is represented by Larry Klayman, a long-time activist on the right and the founder of Judicial Watch. According to the article, they are apparently claiming that the recall petition "violated the free speech, equal protection and due process rights of the majority of county voters who re-elected the sheriff." Note this:
“There are no valid reasons for this recall petition,” Klayman said. “Nothing happened between the day of the election, the swearing-in of Sheriff Arpaio and this recall petition.”I have no idea what the legal basis of this claim could possibly be, but it will clearly be thrown out. Arizona is a political recall state and the recall can be launched for strictly political reasons. Additionally, we should point out that in the Russell Pearce recall, the Arizona Supreme Court held very strongly that the recall is not a joke. No word on whether the group will try find a way to sue the drafters of the Arizona constitution for including this provision.
There is also a more legitimate question of whether Arpaio can be recalled so soon after being reelected. The state law generally prevents recalls from being launched until 6 months after election. But, Arizona law appears to say that does not hold true for a reelection.
The Arpaio recall is still a long shot. Petitioners are claiming that they have 50,000 signatures already. They need to get 335,000 by May 30. To give you a hint of how that signature requirement stacks up, only three recalls in US history have gotten on the ballot needing more signatures (California Governor, Wisconsin Governor and Lt. Governor). The recall of Arizona Governor Evan Meacham in 1988 (Meacham was impeached) needed only 216,746. Additionally, ancillary evidence suggests Arizona's election commissions find failed signatures more frequently than other states.