Michelle Collins, who was deputy portfolio holder for adults and health, and Jack Hemingway, who held the same role for communities and the environment, have both stepped down.
And Coun Steve Tulley has also resigned his place on Wakefield's Police and Crime Panel after supporting an unsuccessful coup to unseat Coun Box.
It follows the resignation of culture, leisure and sport understudy George Ayre last week, after he moved a 'no confidence' motion in the leader at a Labour group meeting on September 17
Speaking on Tuesday, Coun Tulley said those who had wanted Coun Box to step down "won't roll over and die", despite the no confidence motion being defeated.
At last week's full council meeting, Coun Box, who has led the Labour-run authority since 1998, stated his desire to continue in the role.
Coun Tulley, who represents the South Elmsall and South Kirkby ward, said: "My view is quite clear, that a change of direction and leadership is needed now. I feel it's time for a change.
"A fresh look and a new approach is needed.
"What's happened with children's services is a disgrace and it needs addressing quickly.
"My colleagues and I respect the fact that Coun Box is still leader and we respect the result of the vote.
"But at the same time my view has not changed. He should stand down."
All four councillors will continue to represent their respective wards and remain Labour Party members.
None of the deputy portfolio holder positions have yet been filled and it is understood other senior councillors are considering their positions in light of Coun Box's intention to continue.
In a statement given by the Labour Party, Coun Box said: "The new Cabinet team will be formally approved by Full Council at a meeting on Wednesday October 24.
"I am confident that the councillors who will be taking up these positions will have the knowledge, enthusiasm, experience and dedication needed to serve the district well."