You could likely complete the paperwork from New York pro se and the court may allow you to appear via telephone. However, an attorney in New York State could most likely not complete the paperwork on your behalf and then submit it to the court outside of New York, unless the attorney is licensed to practice in the State of Florida. Several exceptions to this rule may apply, for example, if the New York State attorney has a practice order or some other order permitting him or her to practice on a limited basis.
Before filing out any paperwork or sending anything to the court located outside of New York State, “google” the specific court and call the court and simply ask the court clerk if you can mail the files from New York. There may be an additional process required in Florida.
Hopefully, you will not need to make a long road trip :-).
Again, just to be clear, this is not legal advice. I have no experience practicing outside of New York State, so please consult an attorney for legal advice.