Citing executive privilege, the White House said Obama political adviser David Simas was immune from congressional subpoena.
Issa, R-Calif., slammed a letter from White House Counsel Neil Eggleston, calling it "deeply disturbing."
The hearing was to examine the reopening of the political office, which had been closed in 2011, after the Office of the Special Council found it had violated the Hatch Act that prohibits taxpayer funds from being used to pay government workers for political activity. The office reopened in 2014, and Issa says it had again violated the law.
The White House has denied any evidence of wrongdoing, but Issa said "we should not need a smoking gun" to conduct oversight.
The chairman pointed to a 2008 lawsuit brought by Democrats in which a federal judge determined presidential advisers are "not absolutely immune from congressional process."
Issa's frequent antagonist on the committee, ranking member Elijah Cummings, D-Md., slammed the subpoena, charging Issa with failing to provide the necessary evidence to justify a subpoena.
"We do not simply haul in one of the president's top advisers at will," Cummings said.