In Star Trek Bridge Crew, the USS Aegis has a System Intrusion feature that hacks enemy vessels. On the Aegis, anyone but the captain can use it. Does it exist on the Original Enterprise?
Parts are there: Tactical can scan and some details show up on the right wall. But I haven't found the hacking option at any station.
No, System Intrusion is not available on the Original Enterprise.
You can confirm this by activating the help overlay while playing the game (F1 on Keyboard/Mouse). The help overlay will show you what each of the unlabeled buttons does on the Original Enterprise bridge. While the engineering station has the teleporter controls (which were on the other half of the Systems Intrusion display on the Aegis), System Intrusion is not present on any of the stations.
Other players on Reddit have noted the lack of System Intrusion on the Original Enterprise.
