You can easily check this.

First, you may not have a problem at all. From your post, I infer that you haven't taken any pictures yet and instead tried to judge depth of field through your viewfinder. The viewfinder of your camera always shows you the image with the aperture fully open, and therefore what you describe is expected. If you want to see the image with the actual depth of field, push the "DOF preview" or "stop-down" button on your camera. It should be located left of the camera-side lens mount. This procedure is described on page 18 of the camera's manual: AE-1 Manual. You will notice how your viewer image also becomes darker when stopped down.

If this does not work, something might actually be wrong. In this case, the next step is to find out if the camera or the lens is responsible.

To do that, remove the lens from the camera, close the diaphragm to f/22 and look at the lens bayonet.

The lever on the top left in this picture is the one that controls the diaphragm. Try to gently move it counter-clockwise. There should be the kind of resistance you expect from a small spring, like in a ballpen, and the diaphram should open. When released, it should immediately snap back into its inital position.

The first 20 seconds of this video show how it should look like: https://www.youtube.com/watch?v=vlVoxAnvpSQ

If the lever is blocked, the diaphragm doesn't follow the lever's movement, or it feels like you have to overcome considerable friction to move it, then something is likely wrong with the lens. Otherwise, the problem is likely with the camera.