How do I remove a member from my LLC?

Start with the operating agreement, because it is the first place both states look. If it has a buyout clause, an expulsion clause, or a withdrawal procedure, follow it exactly.

If it says nothing, the state decides, and Florida and Delaware point opposite ways. Florida lets a member withdraw at any time, but if that breaks an express term of the operating agreement the withdrawal is wrongful and the member can owe the company and the other members damages. Florida also lists when a member can be pushed out: when the agreement expels them, when the other members unanimously expel them in listed cases, such as when it has become unlawful to run the business with that person or they have transferred their entire interest, or when a court expels them for harming the company or materially breaking the agreement.

Delaware goes the other way. A member may resign only at the times and on the events the agreement specifies, and unless the agreement provides otherwise, may not resign before the company is dissolved and wound up. A member also stops being a member when they assign their whole interest. So in Delaware, a member who wants out of an agreement that is silent usually has to be bought out, and that is the best reason to write the exit terms before anyone needs them.

The cleanest exit in either state is a signed buyout: the company or the remaining members buy the departing member's interest, and everyone signs an amendment to the operating agreement showing who owns what afterwards. Update the state record if the member was named in it. If they were the LLC's responsible party with the IRS, report that on Form 8822-B within 60 days. And an LLC that goes from two members to one is taxed differently from then on, so talk to whoever prepares your taxes before the buyout closes.