Asteroids that have orbits smaller than the Earth's are
Atira asteroids (named after the first confirmed asteroid of this group,
163693 Atira). Asteroids that have orbits smaller than Venus' are
Vatira asteroids (Atira, but starting with a V! Fun.) And those that have orbits smaller than
Mercury are called Vulcanoid asteroids -- or Vulcanoids, for short.
These are currently entirely hypothetical. There is a theoretically stable orbital zone between 0.08 and 0.21 astronomical units from the Sun, and there's not much there to bother an asteroid that hits that groove, but the sun is a Very Big Gravity Well, so hitting that groove is pretty hard. Any that do exist are likely to be under 6 kilometres (3.7 mi) in diameter, and larger than 1 kilometre (0.62 mi). There is some debate as to whether or not any asteroids surviving from the formation of the solar system are likely to remain, as 4.5 billion years of the Yarkovsky effect can do some damage, and we don't see quite the sort of craters we'd expect to see on Mercury if a sizable population had existed and were slowly being collected by their nearest neighbor. However, we don't actually have very good data on that area of the solar system, because the sun is very bright and hard to look at.
The Vulcanoids are named after Vulcan, a planet previously predicted to exist orbiting between the sun and Mercury, based on orbital perturbations and a scant few visual observations. It turns out that Vulcan does not exist, but astronomers love their mythology.