Only a day or two ago there was a woman from Devon on a phone-in on Radio 4 who makes such bird boxes with her husband. Business is a bit slow so might be worth a search for 'devon bird webcam'.
I've no personal knowledge of a bird box camera but do know that on Amazon and elsewhere there are wireless webcams at 45-55 quid (but you'd need an external weather-proof mains power supply, too, which may need an electrician to install, at somewhat higher cost than the camera).
Just looked at search results (mostly for webcams to look at!) and then hit the Shopping button. Seems nearly all sellers of kits and complete bird boxes are on Amazon - do a search for 'bird cam' and you'll see loads of prices.
Good luck with the hunt, there are around half a dozen different things on offer (I did't look further down to see how many more).
(The phone in was about firms evading tax, and the lady who rang sells via Amazon so was asking that those thinking of boycotting Amazon consider which items are from small businesses and which from Amazon itself, before their livelihood is washed away.)