I use Coda2 because it integrates the IDE, terminal tabs, browser tabs, and my (s)FTP file management all in one fullscreened window. Having my entire workflow contained within one app keeps me focused!
But reading your description, I'm left slightly envious and pondering how vim could be integrated better with a browser with web development in mind specifically. I'm not sure what it'd be but it gets me thinking :)
I switch between the two easily enough but I often have to click somewhere to activate the browser part or DevTools. Switching between tabs in devtools is also inconvenient. So that makes me think of a custom app with vim bindings integrating vim, the browser itself and its developer tools... Sadly I'm afraid that would be a huge amount of work in languages I don't know to get that going. Still, thanks for making me think about it!