There must exist a random variable with certain given law?

In general no, but if they are both Polish spaces and the cardinality (either finite, countable or continuous) of $\Omega$ is greater than that of $E$, each endowed with their Borel $\sigma$-algebra and a probability measure, then yes (Kuratowski's theorem)

Consider cases where $\Omega$ is finite and $\mu$ takes too many values...