People ask this question a lot on Avvo. Wired Magazine published an article pertaining to the online-mugshot industry in August, 2011. See link below. It doesn't mention mugshots.com but rather florida.arrests.com, but it's worth reading if you are feeling victimized because your mugshot has been published online, in order to see how other people have dealt effectively with that situation.
Not legal advice as I don't practice law in Florida. It's just my two cents on the facts you describe in light of general principles of law. If you need legal advice, please consult a lawyer who holds Florida licensure. I.e., not me.
The operation of mugshot.com (or mugshots.com) is likely protected by the First Amendment, since the information contained on the site is public record. These sites will often charge to remove the material from their site. While this practice upsets those affected, it is again probably legal since the site can make decisions as to which mugshots to include, and which to remove. This is simply the cost of living in the Information Age, when public information is readily available.
Disclaimer: The foregoing does not constitute legal advice or form an attorney/client relationship. Please contact an attorney for formal legal advice on any specific matter.
Attorney Taylor has provided a useful link. Your mugshot is public information. I think it is a tremendous abuse that these sites have found a way to profit from this fact. Freedom of the press is a very strong protection. In our small town, every arrest, conviction, fine, and dismissal is on the 8:00 AM news and in the paper. Mugshots are often published in the news here. It will be very difficult for you to prevail, if possible at all.
We do not have a client/attorney relationship until you make an appointment, we discuss your case face to face, I accept a retainer, and we explictly agree to enter into representation.