Why is it not possible to have screen readers in all systems? Screen readers are not browser add-ons, they are operating system add-ons, what operating systems are you targeting? In addition to the ones Jeanne mentioned,
NVDA is a free screen reader for Windows, and
Orca is a free screen reader for Linux systems. I understand that MacOS, iOS and Android have built in screen readers too, though I haven't tried them myself.