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It would need to be approved by the court to be legal. While you can take the child with just a notarized letter from mom if anyone calls you on it such as the school DMV or doctor you would have to prove you either had custody or guardianship over the child.
For this reason it makes it easier to get it done now then to wait until someone requests it. A lawyer can draw up the forms quickly and submit them to the courts for approval. If no one is contesting the guardianship it is likely your request would be approved as long as it is a safe home environment and there is no reason why it would not be in the best interest of the child.
If you would rather go without a lawyer you can go to your local law library and pick up guardianship forms. Sometimes these are also available on line to be purchased as well. Fill them forms out and submit them to the clerk of courts office with the filing fee's and the clerk will get a hearing set up where you can plead your case for guardianship or custody.
I wish you the best,