One of the Passiflora species. Passion Flower.

Any idea if there is a way for me to cut some off and regrow it somewhere else?

Looks like the native Passiflora incarnata. If it is like most other Passiflora spp. it can be propagated by cuttings. Can be a little challenging. Rooting hormone and a chamber or clear plastic bag to maintain humidity will increase chances of success. They can take several weeks to over a month to establish roots.

If it's anything like Passiflora caerulaea there should be shoots coming up around the base which you can dig out and transplant.

google : Passiflora incarnata propagation

for specifics ...

the key.. is using the latin name.. as a search term ...

ken

In FL, propagating should be as easy as trimming off a substantially sized piece, sticking it in a hole in the ground, fill hole. Water if it doesn't rain for a few days. Temporary shade like a lawn chair would probably be helpful for the first few days.