Pubget does not collect your institutional login information. It fact, for security purposes, it's set up so it can't even "see" your information. It just opens a connection between you and your institution and then steps discreetly out of the way. For this to work, your browser has to be set to accept all cookies. (If it's not, you'll probably get a warning about it when you first log in to Pubget.)
To do this on Safari, go to the menu bar, click on Safari > Preferences..., click on the Security tab, and where it says "Accept Cookies:", click "Always". Then close the Preferences window.
On Internet Explorer, go to Tools > Internet Options and click on the Privacy tab. Move the slider bar down to the very bottom setting (the widow will say "Accept All Cookies"). Then in the lower right-hand corner of the Internet Options window, click "Apply", and then click "OK".
On Firefox (PC only), go to Firefox > Preferences..., click the Privacy tab, and under "Cookies" click "Accept cookies from sites" and set "Keep until:" to "they expire."