WASHINGTON—In his three years in office, President Trump has overseen an escalation in the U.S. military’s deployment of cyber weapons as a way to engage forcefully with adversaries around the globe without provoking all-out war.

But the White House hasn’t made the same effort to inform Congress and the public about its rules for using these new cyber capabilities, lawmakers and experts say, prompting a bipartisan push to demand more transparency from the administration.

President...