The Church...

Although my character is somewhat attracted to the darker elements of Fallen London, I cannot help but admire those plucky types in the Church.
I have shared some pleasant cups of tea with the Vicar, who is always available for a warm beverage in my lodgings - but - I wonder - are there any other opportunities to hobnob with the holy men?
While I’ve unlocked various storylines with other factions I haven’t had much luck with the vicars and bishops - are there any plot lines to look forward to?

There are a number of storylets down the road where you’re given an opportunity to aid the Church. Generally anything that raises your “Connection: Church” quality, but there’s also the Theological Husbandry story at the end of the Labyrinth of Tigers, where you get to breed beasts with which to assault Hell.

As Urthdigger said there are bits here and there. I think they’re one of the better represented factions in the various storylines of the game. There are a couple of things that are fate locked. If you’re pursing your Heart’s Desire you’ll have a fun bit with the Bishop.

Of course, if you wish to aid them directly - well, if the powers of Fate smile upon you, you might find a way to interfere with the Soul Trade…

Some of my favorite ways to help the church are delivering Hounds of Heaven to the Bishop of Southwark (Theological Husbandry storyline) or turning in souls and contracts to the CVR (Soul Trade storyline).