Spring-loaded holsters kept on the arms are one of the oldest gimmicks in action movies. Whether used by card sharps in an old Western, by the Mariachi in "Desperado," or even by a sociopath in "Taxi Driver," these holsters are often used as a game-changing plot device. While you can make a spring-loaded holster, make sure that you check with all applicable laws if you're attaching a genuine weapon to it.
Fit a bracer to your forearm. It should extend from the wrist up to three quarters of the way to your elbow, and it needs to be made of a thick material such as saddle leather, or costume leather with backing. The bracer can reach around your entire forearm, or it can cover half of it, held in place with straps around the arm. Essentially, the bracer is a platform that you'll be attaching your holster mechanism to, so it needs to fit your arm comfortably.
Attach a sliding track to the bracer. These tracks can be bought at any hardware store, and they're used for sliding drawers. Use rivets in a line to attach the tracks to your bracer. Ideally, use a track that locks in place when fully extended. Before attaching the track, open it experimentally to see which end is the open end. Ideally, when fully open, you want the track to extend down to your hand and lock in place so that the prop on the end is within easy grasp.
Attach a spring to the track. There are two ways you can do this. Either place the spring so that it's stretched out while the track is "closed" and out of sight along your forearm so that the spring pulls the track forward and locks it in place, or alternatively place the spring in the track so that when the track is "closed" the spring is attempting to push the track open to snap one end out to your hand. Experiment with different spring sizes and strengths until you find the one you like.
Install a mechanism to keep the track closed. When you put the track in the same position it would be in to close a drawer, the spring will be trying to pull or push the track so that it's locked open. You need to install something that holds the track closed. This could be a small metal lip that fits over the combined end of the track so that it can't open, or even a magnet that is stronger than the spring power which keeps the track from opening up.
Install your trigger. The trigger needs to remove the blocking mechanism so that the track can extend. For instance, if you have a small, metal plate over the end of the tracks that's holding them in place, that plate could be attached to a metal rod on a hinge. If you flick your wrist, or pull on a hidden string so that the rod pulls the plate out of the way, then the track will spring open. Put the entire assembly on your forearm and try it out to make sure the track opens with the spring's assistance.
Attach your prop to the end of the track that will be shooting out to your hand by mounting it in place with screws. It is illegal to attach a gun permanently to one of these spring holsters unless you have a license for a pistol with a stock, but it is not illegal if the gun is not permanently attached. It is illegal to attach a knife to one of these and carry it concealed if you don't have the license to do so. However, if you are using this assembly for a costume or prop piece, as many cosplayers (those who dress up as anime and comic book characters) and film makers do, then you will be breaking no laws at all.