I am not a business lawyer but I think the answer is yes and you should register with the NY Secretary of State's Office and get authorization for the LLC to do business here. I suggest you retain a NY real estate lawyer.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 385-8015 or via email at Eric@RothsteinLawNY.com. The above answer is for informational purposes only and not meant as legal advice.
You *can* use a Florida LLC to hold title to real property in New York for investment purposes, and in fact, it may be convenient or appropriate for you to do so (rather than choosing an entity formed in some other jurisdiction). However, if you already have an existing LLC that has assets, I don't recommend using that same entity to take title to the investment property in New York for at least two reasons. First, the assets already in the name of the LLC will be exposed to the real estate liabilities, so if the real estate business fails for some reason, your LLC's other assets will be potentially subject to forfeiture to satisfy the real estate business debts. And second, many commercial real estate lenders require the formation of "special purpose entities" (also called "single purpose entities" or SPEs), where the SPE holds no assets other than title to the real property. A different entity typically is used to manage the property. In fact, the ownership structure of the SPE is often a bit more complex, with a division between active and passive investors, and the tax and control implications of these corporate structuring problems can be extremely complicated and difficult even for experienced real estate attorneys, let alone for a non-attorney. You have an added problem that you are a foreign national and potentially will be looking for a way to avoid being taxed on the property's earnings. I don't focus my practice on tax law, but I know enough to tell you that the "phantom tax" issues that arise for partnerships and other investment vehicles in the context of deploying foreign investment monies domestically can be extremely challenging.
As the issues flagged above hopefully indicate, a legal mistake here could mean the difference between your investment being profitable or not. I strongly recommend discussing your situation in detail with an experienced commercial real estate transactions attorney who is comfortable advising on the corporate, tax, and other issues unique to foreign investors. It may be worthwhile to use an empty entity that you have formed in Florida for this purpose, but if you already have an existing business, my guess is that such an attorney would likewise advise you not to use that business to take title to the real estate.
Best of luck with your investment opportunity!