The line judge Sarah Thomas got hit during Green Bay Packers vs Minnesota Vikings game in week 16 of 2016 season. She was taken off the field for further evaluation. So given that she was not on the sideline and the game continued with no stoppage either a substitute referee took her place, or they moved on with one less line judge. But I doubt the latter happened. So are there substitute referees in the NFL (similar to 4th official in soccer) and what are the rules regarding them?
Section 1 - Officials
Article 1. Game Officials
The game shall be played under the supervision of seven officials: the Referee, Umpire, Head Linesman, Line Judge, Field Judge, Side Judge, and Back Judge. In the absence of seven officials, the crew is to be rearranged according to the remaining members of the crew.
Sarah Thomas is on Referee Jeff Triplette's crew. (Official source, Alternate source) Official crews do not travel with an alternate official. If one official has to leave the game, the crew shifts from 7-Man Officiating Mechanics to 6-Man Mechanics, essentially combining the Back Judge and Side Judge positions. There is also a 5-Man Mechanics protocol, if need be. Any less than that, and I would guess that they'd have to call the game off.
According to the Football Zebras blog, Sarah Thomas returned later in the game.