Does the orbit of the sun (and other stars within the Milky way) around the galaxy result directly from radial gravitational force toward the objects in the center of the galaxy, or is there additional forces at play?
I would say the answer to your question is yes and no. Yes that the orbit of our Sun in the Milky Way is due solely to gravitational forces as you suggest, but no in that they are not all radial towards the center of the galaxy.
The motion of the Sun around the center of our galaxy can be broken into two motions. The first is an "azimuthal" orbital motion around the center of the galaxy due to radial gravitational forces from mass interior to our orbit (be it stars, gas/dust, dark matter, dragons, etc.). The other type of motion is an oscillation into and out of the plane of the milky way. This is also caused by gravitational forces, albeit not radial forces from the center of the milky way. As the sun moves "above" the plane of the Milky Way, there is more mass below than above and it gets pulled "down". When it is in the plane it has residual energy from "falling down" and continues to fall "below" the plane where it then gets pulled back "up". This type of oscillatory motion is constant and analogous to a mass on a spring.
Stars in the Milky Way orbit the Milky Way because of all of the mass that comprises the Milky Way. That other stuff includes stars, other ordinary matter such as gas clouds, and perhaps dark matter. I used "perhaps" because dark matter has not yet been confirmed.
General relativity does not explain the orbits of stars about a galaxy with the assumption that the only form of mass is ordinary matter. This leads to two possibilities: Either general relativity (and by extension, Newtonian gravity) is somehow wrong at galactic scales, or mass comes in flavors other than ordinary matter. The former leads to alternative formulations of gravitation such as MOND. The latter leads to alternative forms of mass such as dark matter. Even though the existence of dark matter is yet to be confirmed, the vast majority of astrophysicists lean toward the latter assumption, that ordinary matter is not the only form of matter.