Puckett stepped down after 11 years because he no longer wanted to undertake the travel needed for the job, Michael Page said on Monday.
Bracey, who previously worked in private equity and investment banking, was one of the team that led Ocado's flotation in 2010.
Ocado said it had started a search for Bracey's successor and it also announced changes across its top management team, giving chief executive Tim Steiner greater freedom to drive the retailer's strategy.
It said Jason Gissing, one of the co-founders of the business and current executive director, would become commercial director, taking charge of Ocado's retail activities, including buying, supplier and customer relationships, marketing and brand development.
Mark Richardson, currently head of technology, would move to the new role of operations director, it added.
Ocado's shares have had a rocky ride since listing at 180 pence in July 2010 - they closed at 88 pence on Friday - but Bracey said his confidence in the group's strategy remained undiminished.
"I am certain that Ocado will demonstrate what we all believe - that it is a business with great prospects and an exciting growth trajectory," he said.
Michael Page's shares fell 0.9 per cent, while Ocado was down three per cent.
Earlier this month Ocado, which issued a profit warning in December, posted solid Christmas trading.