Trying to give an object a verb that unlocks two currently locked exits. When I go to click on what I want the script to lock/unlock, it will not show any exits, locked or not, that exist. I'm not sure what I'm doing wrong.

This is on web browser.

It sounds like you need to give the exits a name. After creating an exit it has no default name.

Go to the room you want to unlock exits from.
Go to the exits tab on it.
Click on the exit you want to edit
Select edit from where create would've been.
Give the exit a name.
You can also change if the exit is visible or unlocked initially here.