Downward force applied on a wedge

When you apply a vertical force downwards onto a wedge, let's say by your finger, the wedge pushes back. This normal force is perpendicular to the surface of the wedge and you feel it on your finger.

By Newton's third law, your finger applies an equal and opposite reaction perpendicular to the incline downwards. This downwards force certainly has a horizontal component. This moves the wedge horizontally.

Imagine you dropped a ball onto the wedge, it would bounce off at an angle and receive a horizontal impulse. The wedge then must receive a horizontal impulse in the other direction to ensure conservation of momentum.

Similarly for a weight of mass $m$ sliding down a wedge, it would receive a horizontal component of velocity, the wedge then gets pushed in the other direction. The horizontal force is from a component of the normal contact force. The energy comes from a loss of potential energy of the weight in the earth's gravitational field.

Indirectly the horizontal force is coming from the vertical force of gravity, however there are two opposite horizontal forces, so the total horizontal force is still zero.

The normal contact force is $mg costheta$ between the mass and the slope, where $theta$ is the angle between the slope and the horizontal. The horizontal component of this normal contact force is

$mg costheta sintheta$

A stationary vertical force e.g. applied by a person at one place, is realistically hard to do. The force would either slide down the slope, hence be moved sideways, and be like the sliding mass above - or if held at one place, this would be done by also unknowingly applying a force parallel to the slope, to stop the slide. This extra force would have a horizontal component that could move the wedge horizontally.

As suggested by the other two answers, you can maintain a downward force on the slanted surface of a wedge only if the slanted surface can maintain a static friction force. Then the resultant of the friction and the normal forces at the slanted surface can be upward.