If something that is moving at constant velocity has no net force acting on it, how come it is able to move other objects?

The zero force related to zero acceleration is not a property of the object, it is a statement about the forces acting on the body. That is your title should not read "has no force" but "is subject to no net force".

If a body has a non-zero, but constant, velocity then we know that the total of all the forces applied to it is zero (from Newton's Laws).

We also know that is has non-zero "momentum", and when it collides with another object some (or all) of that momentum can be transferred to the other object. During the collision the body is subjected to new forces and the net force is no longer zero meaning that it will accelerate.

Here's a slightly different but equivalent way to think about it.

Forces describe interactions between two objects. If two objects are interacting, they exert forces on each other. If two objects are not interacting, they do not exert forces on each other. Thus, an object doesn't "carry around" a force with it. A force is not a property of an object, just as dmckee explains. Instead, we describe interactions between two objects using the more-abstract concept of force.

In your block-hits-other-block scenario, it's tempting to ask where did the force come from if colliding object had $F_\text{net}=0$? But when forces are viewed as interactions, it becomes more apparent that the force didn't come from anywhere within one of the objects. There simply wasn't an interaction before they collided, so we wouldn't ascribe the existence of a force force.