The Sun and planets basically do form a disk. The fact that the solar system as a whole is also moving through the galaxy is not relevant to its shape, basically because any external forces (like the tidal force, a difference in force between radially separated points on orbiting object) are not strong enough to affect that shape.

My understanding is that the solar system formed out of a volume of gas and dust. Over time, gravity caused that volume to collapse inward; this in turn led to a bulging in the plane perpendicular to the axis of rotation. This is the same reason that the Earth bulges at the equator. But since a volume of gas and dust is rather less dense than Earth, nothing opposes gravity condensing it further. So it gradually flattened into a thick disk.

Wikipedia’s “Protoplanetary disk” page explains this process with far more fidelity than I ever could. My point is, there’s a good reason the solar system is a disk: it was formed that way to begin with.