I heard that the computing power of ASIC mining machines is equivalent to thousands of GPUs. If you use it to run VanitySearch, it will be very fast.
Can the VanitySearch code be run on ASIC mining machines?
It depends on what you are looking for. If you are actually talking about raw power GPUs win in that aspect but the difference is in what they are built to handle GPUs are better at handling multiple tasks at a time unlike WSIC wares. Mining and vanity gen is not a multitasking software so GPU's raw power will be useless in that aspect.
ASICs are basically built to handle a single task efficiently and since they focus on one task at a time they have the upper hand in this situation over GPUs.