FFXIV: reason the Grand Companies were formed
Very very doubtful actually, as far as the game's lore goes - the whole reason the Grand Companies were formed in the first place was due to the lead up to the Calamity (back in the original version of FFXIV), Gridania, Limsa Lominsa and Ul'dah actually lacked permanent standing armies, only having token militias used for internal peacekeeping (hence why adventurers were relied on for things like monster culling), and were most definitely not in a position to defend against anything, let alone Nael van Darnus and the VIIth Garlean Legion.
Suddenly with the Garlean forces mobilizing for a full on assault and Dalamud looming bloated and bloody overhead like the Sword of Damocles, the fledgling Alliance needed to raise manpower fast as well as providing a quick way of pooling resources together in order to present a coherent front against the Garlean Empire. Thus the Grand Companies were established out of sheer necessity, to oversee the recruitment of adventurers as a volunteer military to defend against the Garlean invasion.
Ishgard however, already has a full standing army and they always have had one (led by the Temple Knights and the Order of the Dragoon), having always been at various stages of war with Nidhogg and his Dravanian Horde for a thousand years. Accordingly, they refused to participate in the Alliance, meaning they were not a part of the establishment of the original Grand Companies - they even have their own volunteer military in the form of the Convictors, meaning even more there is little likelihood of seeing an Isghardian Grand Company I'm afraid.
As for Ala Mhigo, the city has long been conquered by the Garleans, and the remnant forces of Ala Mhigo's royal army are now nothing more than a rag-tag Resistance force that has actually accomplished little in the two plus decades since Theoudic's fall from power and Gaius's annexation of the citystate at best, a good deal of it's troops ended up becoming brigand gangs like the Corpse Brigade that do nothing but make trouble for themselves and their own countrymen and women.
So the Alliance has appeared to fill the void in leading the push towards Ala Mhigo in the hopes of liberating it from the Garleans once and for all (which is of course the whole scenario behind Stormblood after all) - back in the closing cutscenes of 2.55 Pipin mentioned he had been stationed at "the Ala Mhigan Front" for an extended period, proving the Immortal Flames at least have been conducting military operations there for quite some time.
Of course nothing is stopping SE from coming up with some other explanation to explain the establishment of Isghardian and Ala Mhigian Grand Companies. But as the lore stands at the moment, I don't see that ever happening I'm afraid for the reasons mentioned.