WASHINGTON—The House Judiciary Committee voted to authorize subpoenas for the complete version of special counsel Robert Mueller’s report on Russian interference in the 2016 election and alleged obstruction of justice by the president, escalating House Democrats’ battle with the Trump administration.

Attorney General William Barr has said he plans to release a redacted copy of the report by mid-April, but Democrats have called that insufficient, saying Congress should have access to the full unredacted document as soon as...