Can these be made gluten free?

I don't know about substituting a GF flour for the all purpose flour. But the recipe instruction is very similar to Pão de Queijo which is Brazilian cheese puff made with tapioca flour/starch. Here's a recipe for it, http://www.thekitchn.com/how-to-make-po-de-queijo-brazilian-cheese-bread-cooking-lessons-from-the-kitchn-176118
I think you can add in the manchego and chorizo to this recipe.

You can always try to use tapioca flour and experiment. I've never done that so I can't tell you what the result would be.