Upon finishing the Enter Jibanyan chapter of the main story and beginning Protect the Seals, you will unlock the ability to do some sidequests called Requests. As pointed out by Whisper, you will see blue flags on your map, which is the location of the questgiver for these Requests. Talking to them will have them ask a favor of you, which will usually boil down to either getting a specific item, defeating some Yokai or getting a certain Yokai to help with their current situation.

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Requests are marked with a blue flag and will end once you finish the objectives for them.

There are three main criteria that you need to keep an eye out for when doing Requests (and Favors), with the first one being story progression. If you find that a Request isn’t appearing during the correct chapter it is supposed to be available, then you might need to go a little further in the main story of that chapter. The second factor is your Yokai Watch rank, so if you have an opportunity to raise it, then do so. The last factor is the time of day, as certain NPCs might only be available during the day or during the night.