Could a non-Jew run a Judaica store/business? Would this be kosher? What if it's online? Are there any restrictions on converts?
In theory I see no reason per se that would prohibit a non-Jew from buying and selling books, menorahs, and the like, if s/he really wanted to do so. An item on which the seller's word is required that it is ritually okay would be more problematic, but these days your matza, myrtle-branches, and the like come shrink-wrapped and pre-certified, so it's really not about trusting the retailer. (Trusting the retailer with raw mezuzah parchments is an interesting question ...)
The concerns are more likely those of, well, Jews will prefer to do business with Jews all else being equal; Jews will assume a Judaica store is Jewish-owned (and could be off-put otherwise); and there are a whole lot of little things that you're likely to get wrong if you don't really, really know the lifestyle well.