Pulis left Stoke this week.
"The last seven years have been some of the best in our long history and I would like to thank Tony for his huge contribution in this," Coates said in a statement.
"I personally regard him as a great friend and will hugely miss working with him.
"We will now begin the difficult task of searching for a successor to Tony.
"This will be done in private and the club will make no further statement in respect of this search until an appointment is made."