An orbit has an angular momentum that is conserved. That angular momentum vector keeps the orbit at a fixed orientation in inertial space. As a planet orbits the Sun, the local time of the ascending node moves around the clock over that planet's year.
The only way for an orbit to be Sun-synchronous is for there to be a torque applied to the orbit to rotate the angular momentum vector of the orbit in the correct direction once per year.
It turns out rather conveniently that the gravity field of an oblate planet provides just such a torque. But only at the right orbit altitude and inclination. The first coefficient of the spherical harmonic expansion of a non-spherical gravity field is called $J_2$, and results just from the oblateness. The rotation rate of the ascending node is:
$$\dot{\Omega}=-{3\over 2}\ J_2 \left(r_e\over a\left(1-e^2\right) \right)^2\ \ \sqrt{\mu\over a^3}\ \cos{i}$$
where $r_e$ is the equatorial radius of the planet (e.g. Earth). If $J_2$ is large enough, you can select $a$, $e$ (usually chosen to be zero), and $i$ to get a rotation rate that matches the inertial rotation of a Sun-centered frame, about a degree a day. For Earth, the inclinations are around 97° to 101° (mostly polar and retrograde) for altitudes of 400 km to 1400 km respectively.
If $J_2$ is too small, you can't get sufficient torque to rotate the orbit fast enough. That is the case for Venus, whose $J_2$ is about 0.4% that of Earth's. As was pointed out by the mongoose, this lack of oblateness is due to Venus' extremely low rotation rate.
You can get an intuitive feel for where this torque comes from by considering what happens when a retrograde, mostly polar orbit approaches the equator. An equatorial bulge will accelerate the orbit a little causing it to cross the equator sooner than it would if there were no equatorial bulge. If the orbit is retrograde, then the local time below the node crossing will move a little later each orbit. If that amount is the same as the amount by which the inertial frame, e.g. the constellations, moves earlier each orbit, then the orbit is Sun-synchronous.
Due to the eccentricity of the orbit of the planet, as well as the tilt of its rotation axis to the ecliptic, a real "Sun-synchronous" orbit isn't perfect, but it's still pretty good. For Earth, the local time of the ascending node will vary about $\pm 15$ minutes, following the analemma.