If someone cut the cord off it, there is a good chance the monitor doesn't work. People would often cut the cords off dead equipment

When I turn it on, the green light turns on, but I don't see a picture. When I turn it off, I see a faint white square shrinking quickly and then disappearing.

Not sure/don't remember if monitors from this era had those "no input" screens like we have now. Anyone know if there's anything like that on these types of monitors, or would I likely have to put a VGA cable on to test if it works?

From your description it sounds like the monitor might still work. There won't be any kind of on-screen display with an ancient VGA monitor like this one, so what you're seeing is consistent with the video cable not being connected to anything.

If you're lucky the internal end of the VGA cable will have some kind of connector on it that you can salvage. I'd suggest finding a cheap donor VGA cable and cutting one end off. The challenge will be matching up all the wires to whatever internal connector was being used, since you no longer have the original cable to test the pinout!