Oddly, I have visited the Lake Placid area but I know almost nothing about sailing options nearby. There are quite a few lakes around in keeping with the name of the place; the biggest being Lake Istokpoga which I did not visit. Honestly, bass boats and personal watercraft are more in keeping with the local mindset - but I could be wrong.
I am sure you will find more sailors along the west coast on the GOM from Fort Myers to Sarasota about an hours drive from Lake Placid.
I've been sailing in Florida since 1955, but I had to "google" Lake Placid. We're pretty much accustomed to places on the coasts, but I do remember learning to water ski in this area. There's no reason not to sail here, but no need to provision for the bahamas!