If you aren't going to feed a wire into the house then you are talking about a wireless camera. But even a wireless camera needs power. So where will the power come from? There will be that power wire to contend with and you may then have that wire hanging down (hopefully via conduit) to an external power source. Some people have retrofitted wireless cameras with solar panels and battery storage (for night use and cloudy days), but then this is not cheap. That is a lot to deal with even with an external power source. For simplicity what some people do is they put an indoor camera pointing out of a window. That might be an option worth considering.
The wire in an Ethernet camera serves two purposes. It powers the camera and it sends the video into the local network. It is just one wire to contend with that leaves a nice clean install. WiFi cameras are typically more expensive than wired Ethernet cameras. WiFi cameras despite their name do not really offer wireless freedom because of the power requirement, and solving that problem is often either messy or even more expensive (solar and battery). If you are going to need to wire for power, then might as well wire into the network. Unless you conveniently have an external power outlet outside where you need it already. But you asked for cheap and wireless and those two things do not go together unless you sacrifice quality. UPDATE: I just checked and prices have come down for decent WiFi. Still it is easier to find a quality wired camera for a good price than it would be to find the same in a wireless camera.
Whatever you do, do not buy into those proprietary cameras like Nest, Blink, Ring, Arlo....etc. These are proprietary cameras that feed into their proprietary cloud service. You will have a harder time integrating that into typical weather software and personal website. It needs to be a proper IP camera that supports protocols like RTSP, FTP, and ONVIF that don't require service subscriptions.
Some cheaper cameras omit or limit IR (infrared) night vision capability. You don't need a really high megapixel camera to make a good weather cam. Put the money into good IR rather than megapixels.
When you say cheap....what is your budget?
