the sun - Do sungrazing comets leave a field of meteoroids near the Sun?

The SOHO image you attached, provides some evidence, that most of the debris continues the trajectory of the original body.
Most of the debris hence won't replenish the Vulcanoid population.
But due to the disruptive event you get an additional delta-v, which may change the orbit a bit, however insufficient to slow down to a Vulcanoid orbit; compare the velocity of the comet near perihelion with the speed of sound as an order of magnitude estimate of the upper limit of the delta-v for debris.
Dust may either be slowed down in this or in future orbits in the solar corona to end up in the Sun, or blown away by solar radiation pressure, again no Vulcanoids.

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