No cards? That’s pretty difficult for connections, but there are a few other storylet options. You may want to trawl through the bottom of this page:
You can grind Hell, Criminals and Constables by raising Casing and selling information in the Flit. Watchmaker’s Hill has a couple storylets that give Constables CP too, as does the Velocipede Squad (but maybe you locked that).
Public lectures in the Forgotten Quarter give you Revolutionaries, Bohemians, and Society. For Revolutionaries you can also bomb a meeting of financiers with Casing in the Flit, probably faster.
For Docks up to 30 you can get to the end of the Mutton Island carousel and save the ship.
Unless I’m mistaken I think the Benthic & Summerset option under “A colleague with a problem” stays open even if the University is locked, and is the most efficient way to raise those connections anyway.
Church can be raised by breeding a Hound of Heaven, which takes a while since you need a Plated Seal. One of the three Airs-dependent “quick commissions” in Veilgarden also gives a little.
Rubbery Men, probably just Flute Street. Tomb-Colonists… maybe only in the Tomb-Colonies? If only the Widow were actually assisting with an exodus.
For the Widow there are a number of options in Spite, probably the best is attending an audience.
For Urchins, Fun with the Fisher Kings in the Flit: speak softly.
That leaves the Great Game, which is horrible without cards, I think the only option is the Airs-dependent (but around most of the time) option to chat up a tipsy spy in the Singing Mandrake, which gives a whopping 1 CP.