There are digital cameras which will take wifi input from a mobile phone to act as a shutter, as long as you've set up a link between phone and camera. Most of the more recent (last 7 years or more?) will do so, but afaik not necessarily the cheap ones though you can always do research into that.
As for output of video "all the time", you are constrained by the cards used by the camera. Even fairly low quality video on a large - e.g. 64GB - card, you will only get so much continuous video. Again, you need to research this issue. Also bear in mind that European rules limit the length of continuous single shoot video, but I'm not sure why.