I ordered a package from Amazon and I was really happy about it. When I saw that my package hadn’t been delivered I was a bit upset, I called their customer service and the usps and found out that it was delivered to my neighbors who are not home. Should get it or not? Should I leave proof that I am the owner of that mail?
In the US it is generally illegal for you to open somebody else's mailbox. Your best course of action is to contact your neighbor and let them retrieve the package for you.
In practice it might depend on your relationship with your neighbor. If you are good friends, they are unlikely to object or report you to the authorities as they'll be willing to trust that you were just retrieving your mis-delivered package. If you don't know your neighbor, or if you are on poor terms with them, they have no reason to trust your motives and they could reasonably think you were stealing from them or invading their privacy.
As a tangential footnote: yes, people really can face federal charges for tampering with mailboxes. Recently a former city prosecutor and a former police chief for Honolulu were convicted of attempting to frame a relative on federal charges for stealing a mail box. A mistrial was declared on the mailbox theft charges, and in the process, an extensive web of corruption was revealed involving the former prosecutor and former police chief.