Can I make cornbread without flour?
As far as I know, generally the problem with corn based doughs is cohesion.

Cornmeal only mixes tend to be very crumbly and the resulting consistency weak, hence the addition of flour to increase gluten content and help improve agglutination.

That being said, upon a little research there seem to be plenty of flourless recipes around, originating more towards the south of the United States where apparently it is more common to use less flour.

A quick Google search will probably yield lots of recipes for no flour cornbread.

Most just seem to compensate with the inclusion of additional eggs, more sugar, milk or other ingredients to help with cohesiveness.

You may also not adjust at all and simply assume the fact that you will end up with a crumblier consistency and "sandier" texture. It is mostly a matter of personal preference, so if this is not an issue give it a try if you have a chance. Maybe you enjoy the slightly stronger flavor and coarser feel.

In Portugal we have a traditional type of cornbread called Broa which is closer to an actual bread, rather than a quickbread or cake. It is made with cornmeal, combined with rye or other cereals, and the ones considered higher quality generally have less wheat flour. It generally tends to be a heavy bread with dense consistency, dark brown crust with a cracked appearance and a slightly crunchy feel.