Is there a way allow illegal moves in the analysis tool on chess.com?
No.
What are the rules when both players just play on?
If neither player notices and play continues, then there is no penalty or change of any kind.
In FIDE is someone notices then the position is reset to just before the illegal move. The player who made the illegal move is forced to move the same piece if that piece has a legal move.
How would you note this illegal move?
I've never heard of notation for an illegal move... I would just erase it or scratch it out and write the correct move... illegal moves are really rare in tournament play. If no one notices and the game finishes, then, I don't know... maybe write the word illegal so if you're trying to play over the game you know the move wasn't written down incorrectly.
Have you any similar stories of embarrassing mistakes?
I was playing this guy, and I attacked his queen. The queen was in the middle of the board, but no matter where it moved, I was going to win a pawn in one way or another. He picked up his queen to move it, but then noticed one of the tactics, so just held the queen over the board... he did this for about 10 minutes (lol) and finally put it down... but it was an illegal move. He held it so long I'd forgotten where it was originally and I didn't notice. Good thing I won the game anyway, or it would have been really embarassing 
I'm learning how to write notation and decided to play an unformal OTB game to practice writing notation by hand. When I was adding the moves to analyze the game on chess.com I noticed that my opponent made an illegal move half way through the game that neither of us noticed and we played the game to its conclusion.
The move he made was Ne6+ after winning my rook on c8. Embarrassing that neither of us noticed it.
Is there a way allow illegal moves in the analysis tool on chess.com?
What are the rules when both players just play on?
How would you note this illegal move?
Have you any similar stories of embarrassing mistakes?