Cut strips of cardboard off a cereal box and loosen the actions screws, wedge the cardboard down the sides of the action, and tighten it back up - if the horizontal goes away your bedding has slop in it. Even round receivers can have a flat area on the stock and the action moves back and forth. I have a wood Remington stock and the flat spot is really obvious. This is such a common issue there is cardboard in my range bag.
Some shooting styles put side pressure on the pistol grip - if you’re pulling toward you each shot with the trigger hand to line up the crosshairs it will cause weird things because the pressure is different each shot - only move the crosshairs with your shoulder.
Worst case, defects in the barrel can cause it to walk as it warms up. Even custom barrels can get bad steel, factory barrels more so. I recently heard of a new custom barrel that moved point of impact 2 moa in 10 shots. The company that made the barrel replaced it after seeing metal in the chamber was wonky.
There also can be partial contact on one side of the barrel - the barrel vibrates against the tight spot throwing shots sideways different amount.
Loose scope bases also move side to side causing wide groups.
Internal loose scope lenses can move in any direction, but could be left to right.
My experience tells me mechanics of the shooter, mechanics of how the rifles is supported during shooting and parallax issues with eye box alignment. I can't say I've ever experienced/read much about horizontal stringing having to do with action bedding or action screws. Those forces are all in the vertical realm for the most part.
It could just be the load. I say this because I had a Kimber that shot factory Partitions in a 2" horizontal string and would shoot factory Accubonds and factory ELDX at an inch in the same day.
Technique specific:
I have found too much face pressure on the stock will torque the rifle during recoil and can cause horizontal poi shift.
Not maintaining control of the rifle with your firing hand can lead to horizontal poi shift
'slapping' the trigger
Too much 'grip pressure' with your firing hand
Thanks to all. I should have been more precise because I really think it is technique. The variation left to right is minimal at 100 yards with a 338 RUM. About 1/4 in right and one 1/4 in left. Certainly no more than 1/3 of an inch the other 3 shots were essentially the same hole.
I do not know what you call it, but this is the target and it is positioned as it was in the field. It is wider left and right than I normally shoot. So I think it was me just not sure what I was doing, but good advice here.