Whenever you dabble with infinities, intuition must be challenged. They do not behave as finite quantities do.
It is not immediately provable that infinite regress of causation is impossible merely due to the example you provide. An equally valid argument might be that the soldier will fire the bullet after an infinitely long wait. Or, alternatively, the soldier may fire the bullet in a finite amount of time if it is possible for the soldiers and their superiors to communicate with infinite speed. It's even plausible that the soldier could fire the bullet in a finite amount of time with non-infinite communication speeds if one could argue that the order to fire was given infinitely far in the past, before the soldier ever had eyes on their enemy.
These alternatives may seem awkward, but they must be. At the heart of the logic puzzle is an infinite number of mathematically perfect soldiers all of which are incapable of acting without a superior's authorization. By putting such polished gleaming pistons into your machine, you create a physically unrealistic scenario which may be responded to with equally physically unrealistic solutions.
Mathematicians do have tools to explore such infinities, and you may be able to phrase the question in the language of mathematics to come to an answer. For example, if you wish to find a solution using ZF set theory, it is entirely plausible that your representation of "causality" may posit the existence of an infinite regress in a form which is contrary to the Axiom of Regularity (which has much to say about infinite regress). In such a case, you could confirm that such a construct would be impossible in ZF. However, the burden would then be on you as the philosopher to argue why this result has the intended implications in our world.