Yes, you could file a grievance with the Florida Bar for his inaction resulting in a negative outcome on your case. Depending on the type of case that would not necessarily change the outcome.
This is not to be considered legal advice nor does an attorney-client relationship exist.
Schedule an appointment with him in his office just to make certain. If he can't explain the delay to you, and you don't want to have him represent you any more, hire a new lawyer. But, before you do that, make sure your new lawyer is in place and is on-board with filing the grievance. When I hear a prospective client has filed a bar grievance and I am considering taking on the case, that would be a major detrimental factor. And, don't do it after hiring the new attorney without getting advice from the new attorney. now, if the lawyer has missed a critical deadline or fail to timely file suit leaving you with no recourse, he is liable to you. but, you cannot just sit and wait for this to happen or not protect yourself ahead of time. If you do, you will lose the malpractice claim against him.
As my colleague points out, you may have a complaint about professional conduct to the State Bar.
As to a malpractice lawsuit against the attorney for missing a deadline, the obstacles are significant, including demonstrating that you would have won the underlying case.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
Although the attorney handling your case at this time is, unfortunately, not being proactive in this case, that does not mean you have to sit back and watch the deadline pass. I would schedule a meeting with the attorney, and explain your concerns. If you are not satisfied with the answer, you can "fire" the attorney and hire a new one. Of course, I would have already discussed your case with another attorney and ensure that they are willing to take on your case and file by the deadline. This way, you will be able to proceed with your case without missing the deadline. At that point you can determine whether you want to file a bar complaint against the attorney.
I agree with some of the other comments; that is, you should contact the lawyer to see why there has been a delay and if you are not satisfied with the response, seek other counsel. To answer your question, yes, you could file a bar grievance, but you may also have a malpractice action. You don't identify the nature of your claim so I am unable to comment on the applicable statute of limitation.