Correct me if I'm wrong but as far as I know the only way to see your Vanguard/Crucible/Faction rank and progress to the next rank is by visiting said faction in the tower (bar the little progress update you get after completing a mission/crucible match.
It would be super handy if this could be implemented into the options/customisations screen where you can check your bounties and items collected, and where you can see how many marks you currently have. I was thinking a little pop-up menu when you over over your marks would work and be easy to implement.
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The companion app lists ranks both under advisors, and in the tower map section. In the map part tap the vendor in question and it appears at the top of their inventory.