Is there a way to save a zork 1 game in progress? As soon as I close out the tab or browser, it forgets that I saved the game.

Unfortunately not - Zork runs using Parchment which doesn't offer a save game facility as far as I know.

If you want to save your progress you would need to download an interpreter. IFDB has a Download Adviser for Zork here: http://ifdb.tads.org/dladviser?id=4gxk83ja4twckm6j