For browsers, there are extensions that can create SVG screenshots. They work by either copying or inlining CSS into the SVG. They don't work all the time, but worth giving a try first.
For Firefox, see https://addons.mozilla.org/en-US/firefox/addon/svg-screensho..., it worked relatively well for me.
The author also wrote the accompanying library, dom-to-svg: https://github.com/felixfbecker/dom-to-svg
It seems to capture the HTML and convert each element to a similar one on SVG. I'm still not sure how it handles raster images (does it just embed them? Vectorize them somehow?) but it's a really cool project. Thanks for sharing!