Is it possible to determine or estimate the period for Blum-Micali PRG?
Under the question's hypothesis, $x\mapsto g^x\bmod p$ is a permutation of $\mathbb Z_p^*$, that has some characteristics of being random.

If we take a random permutation of a set of size $s$, the length of the cycle iterating that permutation from a random point has length $l$ with probability exactly $1/s$, independent of integer $l$ with $1\le l\le s$. It follows that the cycle has length $l$ or less with probability exactly $l/s$.

This forms an heuristic argument (obvious, as pointed by Thomas in comment) that perhaps it is $\epsilon$-rare that the cycle length of the question's generator is much less than $\epsilon\,p$.

This is however very far from a proof or even a convincing argument. In particular, because $x\mapsto g^x\bmod p$ belongs to a very small and special subset of the permutations of $\mathbb Z_p^*$, with that subset having $(p-1)/2$ elements out of $(p-1)!$ permutations.

By contrast, D.W.'s argument is a valid proof that the cycle length is too large to be found when $p$ is such that the discrete logarithm problem is hard. However that gives a considerably looser safe estimate of the cycle length; perhaps likely at least $2^{100}$ for $2048$-bit randomly-seeded $p$, when the above heuristic argument suggests likely at least $2^{2040}$.