Since Firefox support WebExtensions now (or will soon get it) which are very close to Chrome extensions, it wouldn't be unthinkable for all non-Chrome browsers to use Firefox store instead of the Chrome one. I think Mozilla could even be convinced to open it up as an "open web extensions" store or collaborate with others to build a new one. They certainly have the infrastructure and experience to run one.
Mozilla supports WebExtensions since 57+ afaik. Not sure about this if there are some differences in implementation and API's between Firefox's WebExt and Chrome's.
Uh, Firefox already supports WebExtensions and it is like common knowledge. Mozilla barely has the experience to run a sync service, and even that is a disaster.