I love stone chimneys! I know little about house construction or engineering, but it looks as if there's room for a steel support beam under the chimney, supported in turn by two steel posts on either side of the garage. Regular steel should rust, tying it to the wood siding, and your vines could also trail along it, softening it.

Even if it can be done which I doubt because the load for the stone would have to be transferred to the ground with supports that would run in front of the garage door. Stone is the type of material that visually needs to be anchored to the ground.

you'd lose your garage door, and not sure if you'd have room either side of it for a 'swing door'.