Gameports are simple things. For the most part, the old joysticks didn't actually have any chips inside and didn't communicate with the computer.
Basically, the pots and buttons inside the sticks are wired directly to the pins on the gameport. No real worry of matching protocols, drivers, etc.
Because of that, it will most likely work if you can get a gameport installed.
I have this gameport adapter:
IT DOES NOT SUPPORT ALL FOUR AXIS, so probably only one of your guns would work.
It shows up as a generic 2-axis joystick.
I'd get a $7 sound card that includes a gameport. Make sure the gameport on it supports 2 joysticks.
It will still only have one gameport. The pins for both joysticks are located on the same port.
I've read that some cheap sound cards only support one, but don't really see it mentioned in the specs so it might be trial and error.
Not sure about the chances of windows 7 drivers being available.
XP should be fine.