Closest stable orbit about the sun of an asteroid? What is the closest perihelion an asteroid in stable orbit could have? Is there a limit defined by orbital mechanics or could an object be in stable orbit, only too close that melting or sublimation would prevent the object from surviving?
There's no limit defined by orbital mechanics.  Newton's law of gravitation allows for any perihelion that isn't actually inside the sun.
The Parker Solar probe is in a stable orbit (discounting planned interactions with Venus) with a perhelion of 6.2 million km (at a temperature of 1400⁰C)  But there is no reason that an object couldn't go closer. There is an asteroid 2006HC4 with a perihelion of just 10.2 million km. Its aphelion is beyond the orbit of Mars
Gravitational effects are not the only consideration for stability. E.g. the Yarkovsky effect will tend to deplete the vulcanoid region (Vokrouhlický et al. 2000), YORP effect can spin up asteroids to breakup velocity, radiation pressure removes dust.
However getting into such an orbit is difficult.  If you are a main-belt asteroid, you would need a huge push to get you into a sungrazing orbit.  Comets from the Oort cloud don't need so much of change in velocity (as they are already moving slowly) and so most bodies that we see falling close to the sun are made largely of ice, and are disrupted by the sun's heat.