I am not qualified to answer 508 questions, hopefully Jason Dalrymple sees this and chimes in as he is.
With that said, if adding aria-labels to your buttons will satisfy your 508 issue that is fairly easily done. You would need the HTML name of the button (ex. button123) and then if you use jQuery you can put an action on the show of the page with:
This will add the attribute aria-label=”descriptive name” to the button HTML.
In the attachment I did this to three buttons. I do not have a screen reader so I cannot test.
Caniuse has WAI-ARIA Accessibility features as only being partially supported on all browsers so again, you need someone who is an expert in assistive technology and 508.