Yes, you are allowed to operate a Florida LLC for any legal business in every state in the USA. Under the US Constitution, your Florida LLC is viewed as a legal “person” and other states cannot discriminate against this “person” based on where he/she/it was born or incorporated. This is protected under the full faith and credit clause, equal protection clause, and/or substantive due process clause. Additionally, if your company has a physical place of business, employees, assets, or licensing pertaining to another state, then the state where you operate and have significant contacts may require you register there with a Certificate of Authority, letting the headquarter and significant operations states know you are doing business there. This allows you to bring and defend a lawsuit in that state. This registration does not change the incorporation state, it is only a registry of “foreign” or out-of-state companies, in case someone needs to sue your company in its principal place of business.
Agents and Corporations (IncNow®) offers this “foreign qualification” service to LLCs in every state and can help you register your Florida LLC to do business outside Delaware.