If Mueller is registered today as a Republican, Democrat, or Independent, it is not clear. The last time this was known was in 2001. As he was considered for FBI director by the Bush administration, the Washington Post reported that he was a registered Republican. In the same report, though, it was noted that he is considered by most to be "apolitical," and that he had a history of pushing Democrats for temporary openings, even under a Republican president. "It had never occurred to him that Democrats and the new Republican administration might not mesh," the report from July 2001 reads.
https://www.bustle.com/p/is-robert-mueller-a-democrat-republican-his-appointment-was-lauded-by-both-parties-58693