Only if you like jail. This is a bad idea from start to finish.
You're only allowed to take a bird from the wild if:
you have all the proper falconry licenses and permits
valid Alaska hunting permit
approval from authorities of mews and housing, (they come and check your facilities)
approval from authorities upon notification that you intend to take a bird from the wild
bird has to be of an approved species
Eggs do not count as birds, they fall under illegal possession.
My adviceto you is to do your research on what all falconry entails. what permits and requirements you need, and what type of housing you need. You don't just start by "I think I'll get into falconry, I should go steal a bird." If you think the price of a bird is high, falconry isn't for you. It's not a cheap hobby.
Falcons can hurt you pretty badly if they're trained improperly.
And if you decide to do it anyway, someone is bound to notice that you have a raptor, they're going to ask for your permits which you won't have so you'll be fined and put in jail and the bird will likely be culled.