To unlock the MP-40 in Call of Duty: WW2 you have to level up the airborne division all the way to the end of level 4. The way that you level up the airborne division is to use it in one of your classes during matches. Then enter the headquarters and go see the division prestige guy. Select the airborne division and proceed with the prestige:
This unlocks the MP-40 no matter what your soldier level is.