You can run two accounts simultaneously on the browser. It's just that in my experience the game tends to disconnect you if you aren't tabbed into the game for a certain amount of time. The only workaround I've found is instead of putting them into new tabs you have to put them in a new window so that it won't randomly disconnect.

Hey

Yes you may, http://image.prntscr.com/image/a7f367f78f34461fbe046098491f65bd.png

Once you've put in the login information, It'll allow you to play on two accounts simultaneously