Binding the vertices of one object to the vertices of another object? So I have three separate mesh objects here. There's the gray one on the left, the red one in the middle, and the gray one on the right.

I need to make it so that if I move the object on the left, the corresponding vertices in the middle object with follow the vertices that they are on top of now.

The end result that I am trying for is to be able to move either of the gray objects, and have the red bit in the middle stretch so that it will always be going from the left gray object to the right gray object. I'm hoping there is some way to do this that is easier than writing a complex script.
You could use an armature. A simple rig with two bones is all you need. The left bone will move the left side of the red object and the left outer object while the right bone does the opposite. You could either use weight painting or manually create the vertex groups in the centre object, with the side objects you can parent them to a bone instead of using an armature modifier.