The game wants me to use quotation marks to say something aloud, but I can't for the life of me figure out how to type them.

Did not notice right away you also asked standalone in addition to the 2011 post.

Your profile says "From United States". The quote, ie: ", is right next to the enter key and you use shift with it. And indeed it works in the DOS version of Zork/II/III (that GoG uses). If you are using another version (Apple or Commodore) then you would have to look up how the emulator uses a modern 101key keyboard. On C64 emulators, it is usually shift-2 (which is what it was on the original C64 keyboard layout). On AppleWin emulator it seems to use the modern 101-key keyboard as is. Shift-key next to enter.
Some C64 emulators you can configure to mostly use the keys on a modern keyboard as is, except a few special ones. In which case it would be, you guess it, shift-"key next to enter".
You can look up older systems original keyboard layouts with a simple google image search.

If you use a non-english keyboard layout (or Windows KB layout setting) then I am not sure. You would have to type various keys, and shift-keys, to find out which one does the " if it is not listed on the KB itself.