Where’s the Faith in the Great Game? How do they function on a Trans-Westerosi during this time when kingdoms are going at it? Do they stand neutral/No one has tried to get the High Septon on their side?

This is an excellent question!

The pre-unification Faith of the Seven is a topic that I (and others) find to be a fascinating mystery, because you would expect the Faith to be frequently involved in inter-regional conflict, either as an instigator or a mediator. 

Instead, we have a very brief period where the Faith is used to inspire the Andals during their conquest of the Vale, and a little bit of the same in the Riverlands, but little thereafter. 

This is strange, for a couple reasons. First, the Faith headquartered themselves in Oldtown, and far from remaining aloof from politics, almost immediately we see the first High Septon serving as the regent of Oldtown for twenty years. As I’ve said, this is going to have regional implications.

Second, the Faith established a standing military arm (the Warrior’s Sons and the Poor Fellows) and you don’t do that just to protect itinerant septons, escort pilgrims, and protect septs. You do that because you’re going to be fighting in wars.

However, we do get one example of the Faith involving themselves in the Great Game, which we can extrapolate from to work out a model of their political behavior. As we learn in the Riverlands chapter, the Faith Militant fought for King Humfrey Teague when the Blackwoods rose up against him and then when Arlan III Durrandon invaded the Riverlands to back them up. 

So what can we learn from this?

1. The Faith of the Seven was active in the Great Game. While not a competitor in its own right because it lacked the equivalent of the Papal States, the Faith acted to promote some kings and, presumably, against others. 
2. This activity extended to military intervention. This is something of a risk, because you can imagine a lot of monarchs who would get very nervous about allowing the Faith Militant to operate in their kingdoms if the Faith Militant was going to get involved in internal politics. 
3. The Faith used their influence to shape religious policy specifically. Humfrey Teague didn’t get the Faith’s support just because he was a charming guy, he got it by building “many septs and motherhouses across the riverlands“ and by seeking to “repress the worship of the old gods within his realm.”

So where would we expect to see the Faith getting involved in the Great Game? 

• Well, I would be very surprised if the Faith wasn’t involved in persuading Andals to conquer the Iron Islands (given their bloody reputation on the mainland and their religious differences), or in calling for the punitive raid against Hagon the Heartless (both for his crimes against the Mother and the Shrike’s religious purge). 
• Likewise, I would expect the Faith to have been involved in prolonging the War Across the Water between the North and the Vale once the initial invasions of the North failed, as the best hope for continuing the fight against the pagans. 
• I could also imagine the Faith to try to use the Great Game to crack down on regional religious divides: are the septons in Sunspear getting a bit too “Rhoynish” for the High Septon’s tastes, or is the Most Devout of the Stormlands getting a bit too big for his britches? Well, a war can turn into a crusade very easily.
• And I would definitely imagine that the Great Game would infiltrate into the Faith. Since the Most Devout elect the High Septon, I would imagine that the elections would become very much like the elections to the papacy during the Middle Ages, Renaissance, and Early modern period, where the various nations of Europe would vie to get their candidate on the papal throne as a way to extend their influence and thwart their rivals. Naturally, the Westerlands would have the most cash, the Reach would have the home field advantage, but there’s a lot of votes to be had from the Vale and the Stormlands and the Riverlands and Dorne, so I imagine the competition would get very complicated.