How do you know they were stolen as opposed to not delivered? Make sure you are solving the real problem.
Because someone found the ripped open packaging lying in the road 1/2 mile away. One package (clothing) had tracking showing delivery by post office (which I trust). The other, ironically, was an inexpensive USB camera from Banggood.
As for a camera, you could put a dummy one in the tree with a sign on the mailbox that says "Look up into the tree so I can get a clear image of your face".
Personally I'd try and keep it simple and try a low end dashcam or gopro knockoff triggered by the mailbox being opened.
I considered a dummy camera and may put one up while awaiting installation of a real one. In either case I will put up a small warning sign on the mailbox.
I think if I'm going to go to the trouble of installing one, I'm gonna put one up that not only monitors the mailbox but my adjacent driveway and the road entrance. That way it will be able to record any suspicious activity These will trigger on motion and save to an SD card in a loop fashion.
When I called to report the incident to the Sheriff's office, they asked if any of my neighbor's have video surveillance. They don't. I live in an area with about 30 homes that are several miles from any other homes. Several of the homes are used part time. I have close neighbors but the homes on either side of me are not currently occupied. One neighbor had items stolen from their mail box 2 months ago.
That one looks interesting. So your mailbox is far from your house correct?
Mailbox and road are only about 100 feet or so (30m) from my home.
If the vstarcam is using 12W continuous (probably a bit less), you'll get 45hr life out of a 45Ahr car battery.
That's what the power supply is rated at. I doubt it is using that much. - but that is something I would test. If needed I would set up with small PV to keep battery charged. Could also run power from nearby garage.
Reviews look good: https://www.aliexpress.com/item/Vstarcam-C16S-Full-HD-1080P-IP-Camera-4mm-Lens-IP67-Waterproof-Indoor-Outdoor-Camera-IR-Night/32827834540.html
Thanks for the link. That's a better price!