There are a few ways to handle this, depending on how handy you are with soldering irons and electronic components.
One approach is to reuse the doorbell wires, but disconnect them from the bell transformer (somewhere in the basement? in the utility closet, etc) and connect instead to a PoE (Power over Ethernet) injector. These are commonly available online - used for security cameras, VoIP phones, and other such gadgets. Then use a USB PoE splitter to convert the PoE voltage into 5VDC. You might have to do a little carpentry to fit the PoE splitter in behind your door frame. An example of the latter is here:
Another scheme is to repurpose a Dashcam hardwire kit. These gadgets are DC-to-DC converters, delivering 5VDC to power dashcams. I just bought this one from Amazon to power a dashcam.
It’s quite flexible, in that it will accept either 12VDC or 24VDC as input, and convert it to 5VDC. Some bell transformers produce 20VAC. Another common doorbell voltage is 16VAC. That’s what my house has. If your transformer is 16VAC, you’re in luck. It’s pretty easy to convert the 16VAC into 24VDC (approx), using a half-wave rectifier. That’s nothing more than a series diode (1N4004) and a parallel capacitor (20 microfarad or so).
If your transformer isn’t 16VAC, or if a half-wave rectifier is beyond your DIY skillset, you could replace the bell transformer with an off-the-shelf wall wart power supply, one that produces 12VDC or more (say 15VDC, or as high as 24VDC), at 500 mA or greater. Then use a dashcam kit to convert the 12V (or 24VDC) input into 5VDC for the Wyzecam.
Or, perhaps simpler, a 12 VDC wall wart in the basement, and an automotive cigarette-lighter style USB adaptor hidden behind the door frame.