Can you shoot a 45 caliber pistol on the moon?

Yes, you can. A firearm does not require air to fire- gunpowder contains both a fuel and its own oxidizer within itself (chemically)- now, let's consider some of the results of firing a .45 on the moon- first, due to the vacuum, the lubricating oil will tend to "boil" away, leaving the pistol dry, unlubricated, and likely to "fail to feed" the next cartridge. In extreme cold temperatures, the metal may become brittle, and shatter. Extreme cold would also affect the burning rate of the powder, slowing it down a great deal. The lower gravity and lack of air resistance would make the bullet drop much less than when firing on earth, so you point of aim would need to be higher- and the gloves of your pressure suit will need to be small enough to fit inside the trigger guard. If you do go shooting on the moon, please be sure to clean up your targets and fired brass- we would like to keep the moon tidy.