No. The 2018 shutdown is just a partial shutdown, because Congress has already passed the appropriations for several departments through the end of fiscal year 2019. Defense is one of these departments, so the DoD (including NORAD) is not shut down. NORAD has confirmed this in a tweet:
In the event of a government shutdown, NORAD will continue with its 63-year tradition of NORAD Tracks Santa on Dec. 24. Military personnel who conduct NORAD Tracks Santa are supported by approximately 1,500 volunteers who make the program possible each and every year.
The tweet also suggests NORAD might be able to do this if it was shut down. Military personnel are all exempt from any shutdown, regardless of whether their assigned duties are essential or not. That's because unlike most civilian federal employees, military personnel are entitled to pay based on their status as soldiers instead of hours worked. That, in turn, means military personnel performing assigned duties doesn't result in the government incurring an obligation to them, so there's no obstacle to them working during a shutdown (the same is true for Presidential appointees who are excluded from the normal laws governing leave for federal employees, who are likewise not furloughed). Volunteers don't result in the government incurring obligations (this is different from employees working without pay because volunteers explicitly agree that they are performing services for free), so it's at least conceivable that they could keep running the Santa hotline in a shutdown. I tried to look up the 1995-96 shutdown but Google wasn't getting me anything, so I can't verify if NORAD kept tracking Santa that year.
Again, though, this year NORAD isn't affected by the shutdown at all.