I don't have any experience with iguanas, but you might try bringing you questions to your local county extension.

I do know that they have become yet another introduced pest here in FL.

If it was legal, I'd ask you to catch them and send them to me.

I never thought of those as an issue. It's a first for me. Good luck on what ever guidance you get but hopefully you will not have to kill them.

Can you use screen?

They’re invasive and the official position of the state of Florida is that, as long as it is done humanely, elimination is desireable.

That being said, are you sure it is iguanas eating them? We have lots of iguanas here in Mexico and they frequently come onto my balcony, but I’ve never had them eat anything other than hibiscus.

UF has this advice: https://edis.ifas.ufl.edu/publication/in528

Yes I am sure it's the iguanas. They do not eat orchid leaves, they eat the newly formed buds. They also eat new Croton leaves, and they eat all the lilly leaves in my fish pond.

Do they have a nest? I wonder if it's a tunnel in the earth? If I disturb their nest may be that will get rid of them?

here they often live in spaces in walls; they seem to like an enclosed space to hide in

There maybe a pest removal service that can take them out. The county may offer referrals. Iguanas are highly invasive and nest in a variety of areas. On the sw FL coast they have even managed to enter roofs and damage electrical systems. They will eat many buds and fruit. Other people in Miami have had some success with fencing that is hard for them to climb with small holes blocked off.

Just saw a photo of a group of smallish iguanas that were in the garden dept. of a local Lowe's north of Orlando.

So they're moving in all over and these were most likely traveling in potted plants.

Do they have any natural predators or are we stuck with another invasive species?

Iguana have become a major pest in South Florida, especially eastern Dade, Broward & Palm Beach counties. A number of years back I was down by Dinner Key during the summer and there were hundreds of them in the turf areas and crossing the roads. Here in Martin County they are just starting to be a problem. They decimated my pole beans for 4 plantings in a row.

Iguanas run as fast as a rabbit, climb like a squirrel and swim like a gator so fences and other deterrents don't work. I found the best deterrent is a pellet rifle.

We had loads of the lizards at work in PB County, hired a trapper to no avail. they caught 1 but my housekeeping crew let it out of the trap.

The little ones you see are newly hatched babies. They bury the eggs underground and leave them unattended. Sorry to say but those quiet little ones will grow up to be 4-6 foot lizards with a voracious apatite for your garden. Besides that they love to swim in my pool & make a huge mess on he roof, pool & patio,

Florida Fish & Wildlife Commission, FWC, has said it is legal to kill them year round. I take advantage of that when ever possible.

As far as predator's go, occasionally birds of prey, Bobcats or gators take them. At work we had some Latin fellows in a john boat knock the big lizards out of the trees & into their boat for a later meal. Apparently they are tasty if prepared right.