What digital camera under $150 has the fastest shutter speed and minimal shutter lag?

Can you even buy a reasonably fast camera for that amount of money?
I want a camera that has the capability to capture action well. My Nikon L4 takes several seconds to focus and take the picture, so I nearly always lose the shot.

All cameras take time to auto focus. It's not "shutter lag" - it's auto focus lag. Obviously manual focus would fix the problem, but you won't get that on a $150 camera.

You don't have a chance of finding something that will auto focus that quickly for that amount of money. The easiest workaround is to half depress the shutter release, which will cause the camera to auto focus and lock the focus, once the subject comes into view or the action starts, fully depress the shutter - there will be no delay. This even works with some cheap point and shoot cameras.

Nowadays, nearly all cameras have a shutter speed of 1/1000th of a second or faster, so that doesn't really matter. As for shutter lag, you'll need to spend at least $500 if you want a camera that has very little of it. As long as you're looking at point-and-shoot cameras, there will always be shutter lag no matter what.

If you don't mind the incovenience from using film, $150 will get you a film SLR with a 50mm lens. With some practice with manual focus, you will find that using your eyes and hands to focus is much faster and reliable than relying on a point-and-shoots autofocus system.