The insurgent Libertarian Party will hold the first Florida third party primary for President. From Ballot Access News:

Florida will hold a statewide primary for the Libertarian Party on August 30, for U.S. Senate. The candidates are Augustus Invictus and Paul Stanton. Here is a link to the list of primary candidates so far. The list may grow, because the filing deadline is June 24. Thanks to Rich Moroney for the link.

Florida has not held a statewide primary for any party, other than the Democratic and Republican Parties, since 1920. Although technically all parties nominate by primary in Florida (if they qualify in time for the primary), no primary is actually held unless at least two candidates for the same office file. A few years ago Florida held a Libertarian primary, but it was just for one state house district.

According to the Florida Department of State Division of Elections:

Any registered elector who qualifies for office without party affiliation will have their name placed on the ballot at the general election without party affiliation.

Before collecting any signatures, all candidates (except federal candidates and special district candidates who have not collected contributions and whose only expense is the signature verification fee) must file the Appointment of Campaign Treasurer and Designation of Campaign Depository (Form DS-DE 9) with the filing officer. Each petition must be submitted before noon of the 28th day preceding the first day of the qualifying period for the office sought to the Supervisor of Elections of the county in which such petition was circulated.