We need to know what part of the state you're in, Steve, to make good suggestions. There are things that are great in N FL and terrible in S FL and vice versa.

I'm in South Florida, Palm Beach, about 8 houses West of the Intercoastal Waterway

Hmm. Well, there is certainly a lot of native willow that will grow in S FL, although it prefers damp areas and doesn't get all that tall, but I've never seen the hybrids this far south. Maybe someone else can say more about it.

Tithonia diversifolia! Easy to grow from cuttings. Makes a fast growing, great screen, and supports wildlife without being invasive. I m sure if you post asking for cuttings, people will have lots to share. Just stick 2 foot cuttings directly in the ground, and water daily for 2 weeks and you'll be on your way.

Yeah I'm not sure about 30-40.. that's a bit out of the range of most things considered shrubs. You could try the above bamboo suggestion, a clumping type maybe even one in underground pots for "fast"

as far as natives go that far south there's also non-dwarf varieties of Simpson's stopper but again that's around 20 ft or the aforementioned buttonwood.

Florida privet could work too but it is shorter

if willow hybrids work for your zone then they should be good

if you're handy you can use hog/cattle panels (usually 15' high trellis looking things if you turn them the other way) reinforced with rebar in the ground and grow some sort of evergreen vine maybe a pipevine or passionflower (good for caterpillars). i say hog panels because theyre pretty inexpensive and are how i get my twining&clinging vine trellises

maybe settle for something a little shorter (but that suits what you want in a garden, e.g. the native shrubs) as you save up money for the fence.. but yeah tricky situation best of luck finding something

Edit:

Wax myrtle grow fast and I like the brownish color they often have. I dont think they'll be uniform "shrubby" if they grow to full small tree height, though.

Can you attach anything onto the chain link fence like a Reed Fence ? The chain link would give some support to it. I used a reed fence as a privacy fence while waiting for my hibiscus to grow.

Pitch Apple or sea grape? I don't know if they would grow fast enough for you though.

I'm a bit west and north of you. Southern red cedar, laurel oak, coco plum and wild lime have been the fastest growers in my privacy "buffers". If I lived further inland - I'd plant silver buttonwood too. The wild lime is really thorny. I've read negative reviews of laurel oak, but there was very little damage to them during Irma. Wax myrtle, however, destroyed.

I vote for cocoplum hedges. it is a native plant and you can push it along for quick start and then cutback on water and fertilizer when it reaches your wanted size. As fast grower it will require maintenance and the little black plums (edible) will attract some wildlife you may not want or you may enjoy watching them eat...

^^^ I see those a lot here. I like them but I forgot about them. I also see tons of ixora. The one with dark red blooms is really nice.

Yes, there are all kinds of nifty hedge plants, but don't forget what Steve wants to do: to block a two-story building behind. That severely limits what might work.