So when I boot up Fallout 4 (Windows 7 compatibility mode) I hit play on the launcher and it goes to the splash screen after the video. I hit new, and it goes to the loading screen as it should. After about 15-20 seconds, however, the loading symbol freezes and the game crashes, giving me a little chat bubble in the bottom right corner of the screen saying "Your graphics driver has stopped responding and been successfully recovered". The odd thing is, if I go into the video card settings and disable my video card, the game will actually run, but at 1 FPS. I did this also with a downloaded save file, and the game would play, but still at 1 FPS.

System Specs

Intel Core i5-3230M

8 GB RAM

64-Bit Operating System

Intel HD Graphics 4000

I can link a DxDiag file if needed

Game is not supported to run on most integrated graphics. Even the newer ones that can run it don't run it very well.

Then why can it run without a video card at all?

1 FPS is not running the game. The HD4000 lacks the shader support to run the game. By disabling the card, your bypassing the check mechanism that looks for the proper shaders so the game is allowed to launch. Still does not mean it's going to run, as you have seen.

You need a video card.

Are there any ways to bypass this shader check, and is it in place for Fallout: New Vegas or Skyrim? Because I can play both of those very well with this card, and Fallout 4 is the only game that refuses to work with it

Yes, buy a real video card.

Just because you can get a 4 year old game to run does not mean you can get a new game to run with much higher graphics demands.