If an Amazon driver hits your mailbox the Amazon driver is supposed to give you the number for their dispatch, so that they can compensate you for any damage.
If the Amazon driver hit your mailbox and left without notifying you, then you should contact Amazon's customer service to them through their website or app and report the accident as well as file a claim.
Keep any detailed records of all communications with Amazon and if Amazon is not cooperative, you may also have to file a police report, get the repair estimates and consider contacting your homeowner's insurance or take them to small claims court.
Be sure to take good and clear pictures of the damaged mailbox from multiple angles and if possible notice the Amazon driver's information, the delivery vehicle's details and the date and time the accident happened.
If you got the accident of your mailbox being hit by an Amazon driver on video such as on your security camera, you should also submit that video evidence to Amazon as well.
In my case, I once had an Amazon driver crash into my mailbox during some snow and ice and he apologized to me and game me the number for dispatch.
I was not worried about it myself and just told dispatch and said I would just repair it, but they did offer to pay for it.
They did compensate me for a new mailbox though and I simply installed it myself.
Hiring a handyman or someone to replace your mailbox could cost around $100.00 to $150.00 or so depending on your mailbox and how fancy it is.
Brick type mailboxes can cost $500.00 or $1,000.00 when they are encased in a brick like structure to match your house.
Simple metal or plastic mailboxes are cheaper and can be bought for around $20.00 to $50.00 or so.