Whenever I apply the front brake of my 1960's French 10-speed, the front fork vibrates. Not a safety problem, but very annoying. The fork/handlebar assembly is not loose - first thing I checked. I replaced the front wheel with a new one, problem did not go away. What's next - replace the brake pads? If the problem is the fork itself, I will just have to live with the problem, since I can't imagine that replacing the entire front fork will be simple or cheap. It is a peculiar design - a horizontal tube with the two vertical pieces welded on.
I would try replacing the brake pads and cleaning the rim's braking surface with steel wool. I like KoolStop brake pads. Truing the wheels may also help.