At any moment there will be particles moving in each direction, bouncing on the walls and changing direction. The pressure on the walls of the chamber is due to this bouncing.
Those who go through the hole are not reflected by the wall. Thus though the pressure rapidly equalizes in the entire left chamber, the force on the left-hand-side wall is the pressure times the entire surface, while on the right-hand-side wall, the pressure acts on the surface minus the size of the hole. It thus transmits a non vanishing total force on the left hand side chamber.
As long as the amount of gas in the right hand side chamber is small, during the transient phase, there is no compensating force on the right-hand-side chamber. The entire solid box, during the transient phase, receives a total momentum to the left, and if not connected to any object, (floating in the vacuum ?) it will move to the left.
When pressures are equalized in both chambers, the total force on the solid box as a whole is zero. Not only that, but since the gas has stopped moving as a whole, the right hand side chamber has finally given back to the gas all the momentum the left hand side chamber had received at the beginning, so there is no momentum left, neither in the gas nor in the solid and all macroscopic motion stops.
But the box as a whole will have moved left before stopping, by just the amount needed for the center of mass of the whole thing, box plus gas, to stay put.
If the box is on a table, for instance, friction will resist to its motion, but some momentum is transmitted to the table and from the feet of the table to the ground.