"E" does not open my inventory on Multiplayer

I haven't played on a Minecraft server in years, but I believe that some mini games (such as the ones found on servers) disable the use of the inventory, for what ever reason that is unique to that mini game.

Your controls could be messed up, check your controls

Go into options (in the main menu), then open controls. Scroll down to the inventory section and change your open/close inventory back to E, or reset the whole thing. You also can change it to anything else you want.