Requiring people who get paid to blog about government officials to disclose who paid them and how much.
You can blog about the government all you want with no reports needed.
This law does not require a license to blog, it does not allow the state to approve or disprove content, it does not limit distribution of content in any way, nor does it allow for after the fact silencing.
It says "If someone pays you to write about the government figures in FL, you must disclose who paid you and how much." I have already said that this is not my area of expertise, but there is some case law. Lewis Publishing v. Morgan upheld that the Post Office could require disclosure of editors and owners names before allowing a Newspaper to use the mail. Barber v. Time upheld that the state must balance "Proper public interest and an individual's right to privacy". The legislature has explicitly said that it feels it's in the public's interest to know who is funding non-newspaper, commercial, political speech. In general the USSC has ruled back and forth on the 1A protections of commercial speech vs. just a citizen talking, and while current case law is that commercial speech is mostly protected under the 1A, it does stand apart as open to more regulation than the non-compensated speech of an individual citizen.
The Chilling effect doctrine does exist, and there were several cases in the McCarthy era that specifically held rules forcing commies to register or appear in person weren't constitutional, but chilling effect is not a blanket prohibition on all regulations. The anti-war protesters in the 1970's found that out. That case was about .gov surveillance (a passive chilling effect) rather than a requirement for an someone to act (file a report), so it's not a direct corollary, but it shows that there are exceptions made to the doctrine.
Again, if you have a specific part of this that you think automatically fails USSC scrutiny, I'd be interested in learning more case law, but on the face of it I don't see how requiring disclosure of who paid for a political article is ipso facto a 1A violation.