[mothered]

the package is in my hands, should I wait until midnight USA time to contact amazon?

You're not supposed to know when It arrived, so contact them the next day saying you're still waiting for the delivery.

and live chat or call in?

If you're very confident over the phone, then give them a call.

On the other hand, If you use chat, you may be In luck to communicate with their chat bot who could approve the claim there and then. If not, you'd obviously be dealing with their rep/agent.

Ultimately It's your decision.