WASHINGTON—With President Trump vowing to “fight all the subpoenas,” the Democratic-led House of Representatives is gearing up for a protracted legal battle over its power to enforce demands for documents and testimony from the administration.

Democrats have aggressively issued subpoenas since taking over the House in January, including calling for an unredacted version of special counsel Robert Mueller’s report on Russian interference in the 2016 election and any obstruction of justice by Mr. Trump.

The...