Looked at this gravel screen and its a different design wondering how the shaker box moves sideways directly connected to the small gas engine, How or what mechanical connection would be involved? https://youtu.be/BG8YXpPKhLU?t=36
You can determine the plane of rotation of the engine by noting the pull starter on the front of the engine. This puts the axis of rotation parallel to the grill slots of the sieve.
The engine must have an eccentric plate of some sort, akin to the linkage of an old time steam engine.
There would be a bearing on the eccentric plate where it connects to the link rod. Visible in the video is the other end of the link rod where it attaches to the shaker box. There appears to be a rubber mount in that attach point, but also might have a bearing inside.
Note that the shaker box is mounted on chain, which allows for the side-to-side movement.
With the power and speed displayed in the video, very little translation is needed to accomplish the objective.
A watt's linkage has very good linear properties and is very simple.