I would say no. The Caravan Bond ability allows the Varisian Pilgrim to "use her domain-granted powers on any of these traveling companions as if they were her." A proficiency is not an ability you use, it is something you have. You couldn't pass it to all your friends any more than you could give them all animal companions with the Animal Domain or give them all +10' movement with Travel.
Anything that you actually "use" would be fair game. For example, you could use the Order Inquisition's "Mantle Against Chaos" on any of your chosen allies within 30', or give them the Persistence Inquisitions "Relentless Footing" bonus to speed. Anything with an activation would work, even if it was normally self only.
If Inquisitions work, I wouldn't see why that one wouldn't.
However, the more I look at it, the more I think I was wrong on Inquisitions working in general. The specific text says that Inquisitions they "are like domains," not that they count as domains (even though you can select them in place of domains). So, super strictly, I think you might not be able to use any of them after all.