It will not cause lag for the player.
Spectator mode receives information from the game server, not from the player who you are spectating.
Not even considering the lag it could cause if the player's game client had to stream the game, the biggest reason for this is because it would be unreliable since it would have to deal with the player's network setup in order to send data to the spectators. The player's home network is an unknown variable to Riot so it would take a large effort to make this work, and it would still be unreliable compared to having spectators connect to the game server.
Also, the game server brokers the game state to the players in the game, so logically it follows that it should also broker the game state to spectators too.