WoT says that non-mages were a part of the clergy before there were Grand Clerics, as Grand Clerics was a role designed by the Andrastian Chantry, and the Imperial Chantry is a return to the past (with its theological interpretations, anyway, kept the structure). It was in Towers (the 3rd Age), when a "Black" Divine was established and the schism made, but before that, there were still mages who were part of the clergy in Tevinter that Val Roaodjfasjdl (w/e that orlesian city is named) didn't know about.
With how Tevinter culture (and the Imperial Chantry since it is only made of Tevinters at this time) views magic as a divine gift (a traditional idea even before the Old Gods), it does make sense that the upper levels of the Chantry being for mages, as they seem to be given the responsibility of a divine gift. Whether that same custom extends to Imperial Chantry communities outside Tevinter, it has not been said... since there are none, currently, but the question is whether this is now a doctrine of their faith or just a tradition.
Still not a whole lot of information has been given of either Chantry. This lore is all pretty new, DA isn't that old, and much lore is hidden from us until it becomes relevant.
Currently, what we do know is that non-mages have a glass ceiling in Tevinter, they can't reach the higher offices of Tevinter or the Imperial Chantry, as those are filled by members of Circles, and I don't think a non-mage can apply to a Circle.