There's more to hunting monsters than hacking and slashing.
Sometimes you have to question witnesses, interrogate suspects that might secretly be monsters in disguise, comfort widows and other friends of family of deceased victims so maybe you can coax some information out of them...

The question is, how do you do such things in a game without rolling dice?
We can start by asking how to do such things in real life.
You could gain people's trust. You can do this by showing authority (flashing a badge for example), or by showing the person that you are just like them (like relating some sort of life experience to show that you know what the person is going through.)
You could pressure and intimidate people. This can be done with authority, or it could be done with a show of force, or by exploiting the person's fears and prejudices.
You could insult people. Bring out their insecurities to agitate or soften them up.
You could even use shear repetition. If you keep telling a person the same thing over-and-over, they are sure to believe it eventually.

I'm thinking of using some sort of system of 'cards', where every character has a number of cards like 'authority figure' or 'pity', or 'interesting analogy', and you must spend these cards as a resource to get information out of people.

Perhaps important NPCs will have grids where the GM will put check-marks or X's in squares to indicate what sorts of 'cards' will or will not work on that NPC. These grids will be kept hidden from players. The social mechanic of the game may work a lot like a game of Battleship in that players will have to try out a variety of cards to figure out what works through process of elimination.

Any suggestions on what sorts of 'cards' could be used to get information out of people?

Give the people what they want: flattery, seduction, bribery, respect, a chance to tell their own side, sympathy, commiseration.

Give the people a favor or perk: patronage, a good word, a chance to lok good in the right circumstances

Limited time offer: not just above, but lessening of negative consequences like a fine or prosecution or exposure

Appeal to civic duty, ethnic or professional pride (vs unity, which usually keeps mouths shut)

Coercion: threat of or actual harm (physical, social, emotional, reputational, institutional) to target, family, business, community, special interests

Just watch any doctor cop lawyer show where they question witnesses, and write down the approach used.

I'm no fan of Law & Order, but I might have to watch a couple episodes of that and mine a few ideas from it.
I'll have to include "Guy who continues moving boxes while talking to the police" as an example NPC when I put together my game book.

Kev,
  If I were making this system, I would setup two sets of cards. One would be a set of opposition cards. They would have the NPCs disposition (e.g., "fights authority" or "has something to say") and a set of "social moves" (e.g., "Bluffs", Intimidate", "Impress", etc.).
  And then have the moves and opposition cards have tags that work like the elements on pokemon or JRPGs. Like, maybe "fights authority" is vulnerable to "Flattery" but Resists "Intimidate" or something.

Thanks Dave!
I'm not sure if I want to bring a 'rock, paper, scissors' subsystem into this, but I might give it a try.

I did find a flash of inspiration in your post though.
I wonder what I might be able to do with a simple 'Trust' vs. 'Distrust' binary.
Very simple.
Character sheets may include a list of descriptors such as 'Authority Figure' or 'Average Joe'. First you put a check mark next to each one that applies to you. These are your 'cards'. Then, next to each, you mark down whether your character trusts or distrusts other characters with that descriptor. There will also be a space to write down some details, including special exceptions.
When interacting with NPCs, you 'play a card' each turn, hopefully to gain the NPC's trust. If the NPC trusts you, you will get information. Otherwise, nothing. The NPC may get to play cards against you, perhaps to leech a resource like 'time' or 'sanity' if the NPC is hostile.

The suggestion reminds me of the interaction system for the computer game Republik. You had cards with types of moves that you scripted out using, and learning some facts about the person gave you a chance to learn what might work. The cards were used in more than one round, so you learned what they had to play and what would best work for you of what you had choosen.

Kev,
  I wasn't actually suggesting a RPS mechanic, I just meant that whatever mechanic you use, that certain moves get a bonus versus certain opposition cards. So, if my Personality stat usually reduces your Will stat by 4, if I have an "Flattery" Move and you have a "Vain" Opposition card, maybe it reduces it by 6 instead.
  Sorry, I wasn't clear before.

Oh! Sorry I misunderstood.
That makes sense actually. I'll think about this while I'm at work today and bounce some ideas off you later. Thanks! :)