Graphics Card: Intel(R) HD Graphics
Memory: 2GB
Processor: Intel(R) Pentium(R) CPU B940 @ 2.00GHz

I'm pretty sure I was able to make MoH run on an even older laptop with lower specs but for some reason, when I run it here, it just crashes (all 3 MoH).

Check the game's system requirements from the gamepage:

Minimum system requirements: Windows XP / Windows Vista / Windows 7, 1.8 GHz Processor, 512MB RAM (1 GB recommended), dedicated 3D graphics card compatible with DirectX 7 (compatible with DirectX 9 recommended), 3GB HDD, Mouse, Keyboard.
Compatibility notice: Intel integrated graphics cards, and ATI Radeon X1800 cards, are not supported.

Notice the last line. Intel integrated graphics cards are not supported. I'm not going to say it is impossible to get it running on integrated graphics are there are some posts on here where some users have managed to get it running, but it cannot be guaranteed.

Here is one forum thread that has a fix that works for some.
http://www.gog.com/forum/medal_of_honor_allied_assault_war_chest/how_do_i_get_the_game_running_on_a_laptop_with_an_integrated_card