Assuming that you are currently using a GUI based browser (Chrome, IE etc.), you probably want to switch to using a keyboard based browser (such as Conkeror). To quote the web page:
Conkeror is a keyboard-oriented, highly-customizable,
highly-extensible web browser based on Mozilla XULRunner, written
Emacs and vi. Conkeror features a sophisticated keyboard system,
allowing users to run commands and interact with content in powerful
and novel ways. It is self-documenting, featuring a powerful
interactive help system.
I have added the italics to highlight why it could meet your needs. It works on most OSes, including Windows.
Here is a tutorial on how to copy text, in Conkeror, from the MIT Conkeror Tutorial
Gecko has a feature called caret browsing that gives you a blinking
cursor that you can move and select document text with the keyboard.
To enter caret browsing mode in Conkeror, first press C-M-q to enter
content-buffer-quote-input-mode. All of your keystrokes until you
press ESC will bypass Conkeror's key handling and be passed directly
to Gecko. In content-buffer-quote-input-mode you can enter caret
browsing mode by pressing F7. Move the caret with up, down, left,
right, and so on. Select text by holding shift while moving the caret.
Exit caret browsing mode by pressing F7 again. Press ESC to exit
content-buffer-quote-input-mode. Press M-w to copy the selection to
Hope this helps.