SALEM, Ore.—Law-enforcement officials closed the state Capitol on Saturday, following threats from militia groups that were made while a partisan showdown has halted all activity in the Oregon Senate.

“The State Police Superintendent just informed the Senate president of a credible threat from militia groups coming to the Capitol tomorrow,” read a text sent out to senators on Friday that was shared with The Wall Street Journal. “The Superintendent strongly recommends that no one come to the Capitol.”

Oregon...