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but that's tabbing with the left/right keys instead of the tab key, which is how I tab through a website

ignoring the fact that there's no :active styling in the css, there is nothing to indicate that to change the tab you need to use different keys

I had to first click on one of the tabs before it started letting me use the keyboard.

Yes it is; I can switch between tabs just fine with arrow keys.

I have had a WCAG AA 2.0 audit failure doing this because this approach is not obvious to a sighted keyboard only use that space bar and arrow keys and not tab + enter is the control mechanism. Also a non sighted user will need hidden text before the tabs at a minimum to explain how they work.

Sure it is, I did so before I posted.