If you have unlocked the Sanctuary and had a look around, you probably found that friendly guy at Headquarters offering requests from your allied characters. They reward you with a notable amount of regard and a card from the requester. What is not immmediately obvious: The requirements and rewards are fixed, and requests reset along with your main story (i.e. start at stage 1 again on every new game plus).
I just gathered the information on quests so you can check if the rewards will be worthwile, and if you should clear the map of certain battle types or wait for requests for it. Or just do whatever with it.

The cards you gain from fulfilled requests follow a certain structure.
While the Friendship Gift and Attribute vary per character, their strength does not. Here's a quick list of the common stats:

Welcome to the main part of this guide. Since some details of the rewards were noted above, here's just a list of each character's quests and their specific rewards.
Due to length restriction per section, it's split into 2 parts.

This is a messy list. While I am still working on filling it, feel free to message me about any information or corrections.