The last ability will trigger, at most, once per combat damage step per player. However, if at least one artifact creature you control has first strike and others don't, or if an artifact creature you control has double strike, the ability could trigger twice per combat: once in each combat damage step.

Thopter Spy Network's first ability has an “intervening ‘if' clause.” That means (1) the ability won't trigger at all unless you control an artifact as your upkeep begins, and (2) the ability will do nothing if you don't control an artifact as it resolves.