From the LCS rules book, you can read under point 3.4 Substitutions:
3.4.1 A Team may substitute a Player between Games of a Match. The Team must notify a League Official and have the substitution approved,
no later than 5 minutes after the explosion of the Nexus in the
3.4.2 In the event of an emergency, a Team will be given up to two hours to find an immediate Substitute from their Roster for a Game. If
a replacement cannot be found, the Team will forfeit. League Officials
will determine if an event qualifies as an emergency.
3.4.3 Player substitutions have to result in Teams having eligible Rosters
To me it sounds like you can swap a player mid game, but only if there is a serious emergency.
I am not quite sure how the game you mentioned continued, but a disconnect can happen due to various reasons like a bluescreen, power-outage and so on.