Really, this is just an upscaled space elevator.
Most others have already pointed out that it may be possible from a materials point of view, but I don't think anyone's pointed out what other issues you might have.
Let's look at an Earth-Moon bridge for a more local example:
One problem we have is that the Moon orbits at a different velocity to the Earth's surface - so anchoring it at this end could be a problem. The Moon itself is tidally locked, with the same face always pointing at Earth, but there is still a small wobble during its orbit, so anchoring might be easier there. Oh, and the Moon is receding (slowly) and the Earth's rate of spin is slowing, too (one is a consquence of the other).
Then, we have this giant ribbon of material undergoing gravitational attraction towards the Earth, gravitational attraction towards the Moon, and so it will be under massive tension. Not too mention, the Sun and other planets will have some gravitational influence.
Let's get back to the Sun-star bridge:
As as been mentioned - all stars are in relative motion to each other - but locally, this isn't all that fast.
One problem is that to prevent the bridge from just collapsing into the Sun, its will need to be in orbit around the Sun. I'm pretty sure the maths on this is going to get very complicated for an elongated body (compared to the relative small spherical bodies we're familiar with).
This in itself will setup tension along the bridge (some of the bridge is being pulled toward the Sun, some of the bridge is accelerating away from the Sun as it orbits).
Then, eventually, as the bridge is extruded out towards the other star, we're going to have to account for additional tension of the bridge being pulled in towards that star.
And then there's the gravitational effects of all the other local stars.
I think what all this adds up to is a mathematical nightmare of orbital mechanics...