Besides the orphanage in Riften, I know you can get the beggar girl Lucia in Whiterun. --- Playing - Darksiders 2, Sleeping Dogs, Walking Dead, Spec Ops, GR: FS, Lollipop Chainsaw, ASM, Skyrim, Starhawk, AC: Rev, Asura's Wrath, ME3, MAA
I found a little dude who works all day just so he has enough money to sleep on the floor by the fireplace at night in dawn star at the inn next to the barracks I believe. His Dad was a sailor and when he got sick the crew left him and his son in dawn star. A few days later his dad died. Unfortunately they don't always let him sleep there, and dawn star is wickedly cold, so my argonian wife and I, previously living in wind helm, moved to a house I made in the hold so he wouldn't have to move far.
I tried to adopt from the orphanage but it wouldn't let me
I think all you need to do is have a house with room for children, ( if your house is in a major city you can buy space from a steward) which apparently consists of a bed+ one storage container per child. It also might be possible to fail the background check ("what do you do for a living") although I doubt it.