It was Springtime at President Trump’s Mar-a-Lago club, and the favor-seekers were swarming.

In a gold-adorned ballroom filled with Republican donors, an Indian-born industrialist from Illinois pressed Mr. Trump to tweet about easing immigration rules for highly skilled workers and their children.

“He gave a million dollars,” the president told his guests approvingly, according to a recording of the April 2018 event.

Later that month, in the club’s dining room, the president wandered over to one of its newer members, an Australian cardboard magnate who had brought along a reporter to flaunt his access. Mr. Trump thanked him for taking out a newspaper ad hailing his role in the construction of an Ohio paper mill and box factory, whose grand opening the president would attend.

And in early March, a Tennessee real estate developer who had donated lavishly to the inauguration, and wanted billions in loans from the new administration, met the president at the club and asked him for help.

Mr. Trump waved over his personal lawyer, Michael D. Cohen. “Get it done,” the president said, describing the developer […]