The CBI said the "political vacuum" needs "to be filled at speed to protect people's living standards".
The Institute of Directors warned "what business hates most is uncertainty and instability".
Mr Johnson has said he intends to stay on until autumn as a "caretaker" prime minister.
But it is not clear whether that will happen.
Some MPs and business leaders suggested he should go sooner rather than later to allow a new executive to try to tackle the sharpest rise in the cost of living for 40 years, which millions of UK households are struggling with.
