Now that I have re-read your report, I see that you are thinking that Hyperbole buttons utilize the Emacs button.el library (which has a keymap that supports tabbing through buttons). However, Hyperbole buttons predate that package by about a decade and so have never utilized it. So this is not a bug (either in Emacs or Hyperbole).
I guess what you want are commands that move forward and backward among explicit or named implicit buttons in a buffer which you can then bind to keys that you like. I would suggest you close this report and open a new feature request, describing the full set of behavior you would like. Then we could work on that.