Whose beach chairs as carry on luggage?

You say you are doing only carryon so I’m assuming that you’re bringing a suitcase. Don’t think there’s a beach chair out there that is going to fit the dimensions for a personal item that fits under the seat in front of you but who knows….

Any reason you don’t want to buy inexpensive ones from one of the stores along the beach in Tavira? Any typical metal framed beach chair would not fit the personal item dimensions on an AA economy ticket allowance. You might be able to get some personal items into a bag in between the seat and back of a stadium sized seat and carry it on as your personal item? stadium seat Weight (1lb) and dimensions when opened are included in the linked description.

I've not done this but you should probably expect to get extra interest at security screening due to the metal components of chairs. You could contact TSA, though the final decision rests with the officer inspecting your luggage.
https://www.tsa.gov/travel/security-screening/whatcanibring/sporting-and-camping

A quick google search offered plenty of suggestions, like this pricy option https://www.cliqproducts.com/pages/cliq-products

Thank you all for your interest! Concluded it makes more sense to get them at our destination.

You get one carry on item and a personal item such as a purse size item. So how would you bring a carry on luggage and a beach chair?

You would need to check the beach chair. You would probably just tag it at the check in agent pre TSA and they check it for a fee. But I'd check with American if that's ok. In the old days, sure, but now who knows if they can just toss something like that on to the luggage belt. The cost likely more than purchasing one upon arrival.

The funny thing about this is 30 years ago, our then toddler son loved to watch the stackable washing machine go around at the vacation condo we rented. So we bought a step stool on vacation and it was really a good one, so we checked it as is, and brought it home. Back then there we no fees, and nobody cared that we wanted to check it. Just came off the luggage belt as is. It literally just broke last month. 30 years of use! So, I'm not thinking checking a chair is strange; but the world has changed so much with anything you check or carry on now. Plus the expensive fees.

For a minute I thought it was April fools day.

Good decision to get cheap ones there.

John D, wouldn't taking chairs as your carryon mean you couldn't take anything else, like clothes? I know you're going with another plan, but I think some of us were thrown off by your original plan to go carryon only.

There are camp chairs that will fit in your carryon as they fit in backpacks. Wendy Outdoors on YouTube has one she loves. As comfortable as 'beach chairs', I don't know. Perhaps go to REI or such to see what they have. You can do this if you like the chairs and you want to allocate the room. However, they are likely more expensive than buying one there.