Although this has not been officially confirmed, the Associated Press and major news sources are reporting the name of a suspect. From the NY Times:

Law enforcement officials identified Vance Boelter, 57, as a suspect in the assassination of one Democratic state legislator in Minnesota and the attempted assassination of another on Saturday. The suspect’s identity was confirmed by two law enforcement officials with knowledge of the matter who were not authorized to speak about the ongoing investigation.

State, local and federal police have flooded the Minneapolis suburbs searching for the suspect, and for hours have asked residents near one of the crime scenes to shelter in place. Officials said a gunman impersonating a police officer killed Representative Melissa Hortman and her husband, Mark, on Saturday morning and wounded Senator John A. Hoffman and his wife, Yvette, in separate attacks at the lawmakers’ homes.

[. . .] The targeted shooting of two Democratic Minnesota state lawmakers on Saturday morning came days after the conclusion of an unusually acrimonious legislative session, where tensions over party dominance had turned into a legal dispute, and a narrowly divided statehouse struggled to agree on a two-year budget.

State Representative Melissa Hortman, a top Democrat, and her husband, Mark, were fatally shot inside their home in Brooklyn Park early Saturday morning, in what the governor described as a politically motivated attack.

About the suspect, from The Mirror:

Two people familiar with the matter identified the suspect being sought as 57-year-old Vance Boelter. The people could not publicly discuss details of the ongoing investigation and spoke to The Associated Press on condition of anonymity. Suspect Boelter had a sickening manifesto and connection to Walz.

The New York Post reported Boelter was appointed by Gov. Tim Walz in 2019 to served on the Governor’s Workforce Development Board.

He previously served on the Governor’s Workforce Development Council, appointed in 2016 by then-Governor Mark Dayton.