For a pet crow, you need to acquire a nestling when it has pinfeathers, then raise it by hand.
Technically, you need a federal permit. I don’t know how you get one.
Here is a link with a discussion about owning pet crows.
It is a good discussion about what owning a crow is all about. It also says
Do not attempt to capture and train a native wild crow, especially in the US, where a bizarre set of loopholes in the Migratory Bird Act in several states will do things like allow you to kill as many wild crows as you like between October to May as long as you’re using a shotgun because crows are considered vermin , but will also give you huge fines and even a prison sentence if you find a baby crow and raise it because… they’re protected by the Migratory Birds Act.
Here is another article on why not to have pet crows.
https://pethelpful.com/wildlife/Crows-as-Pets-Is-it-Legal-to-Have-a-Pet-Crow
In the US, all migratory birds are protected by the Migratory Bird Treaty Act. Although crows don’t always migrate (only the ones who live in the coldest places do), every type of crow you might find is listed on the Department of the Interior’s Federal Register of Migratory Birds. So this means that it is indeed illegal to take/possess, export/import, transport, sell/purchase, or trade any crow or any part of a crow (including its feathers) or their eggs or nests, without a permit. It’s also usually illegal to hunt, kill, capture, or take them without a permit.
But what if I find the crow injured and nurse it back to health?
Also from that first link
You be prepared for your chosen companion to be a monumental jackass for his entire life, because virtually all corvids are a$$holes to the genetic level.
Trained or not, their personality is some flavor of ‘jerk’.
Where other creatures have amino acids in their DNA, crows have buttholes.
They’re made of buttholes, spite, sadism, and pure, unadulterated hilarity.
See? Sounds like just the pet for me.