In my opinion, I believe only armigers -- those bearing arms -- should be permitted to register badges in this specific way. They were originally intended to serve as a way to label items and members of a household, since only the armiger themselves would directly bear their actual armorial achievement. The crest from the achievement was often used as a badge in part or whole, which gave rise to the common synecdoche of a "family crest."
However, it's worth noting that the badge phenomenon evolved from a mark of livery to something more akin to the modern-day logo, and often no one bothered to register them anywhere (indeed, often they weren't registered at all. So if someone wants to bear a badge they haven't registered, neither law nor custom should make them feel like they aren't allowed.
This is all my own opinion, of course.
-NRH
