The problem is you'll likely need to pick and choose players until you get a number that likely won't be beaten.
For example, we'd start by listing all NHL players, either active or all, depending on your desired result. Then you'll look for players with more than 16 career goals because you know Andrew Shaw has 8 games at 2 goals which means no one can have more games but fewer goals. Then you'd need to eliminate anyone with fewer Games played than 2x Career goals (ex: 16 GP, 33G) as this would mean they had a hat trick in at least 1 game. You'd then take a list of players with recorded hat tricks and remove them from the list.
This will leave you a giant ass list of NHL players that possibly break the record. Start checking each until you get a number higher than 8 and repeat the steps above.
None of the hockey stats sites I've found keep track of 2 goal games as its an arguably insignificant stat. When announcers are calling a game, they have a MUCH smaller pool of players they could look up stats for and could start looking for this information once something starts to head toward something significant.
In short, you'll likely not find any direct answer to the question and would need to start to pick and choose players to narrow it down enough to have a high confidence in whomever you claim is the one with the highest amount.