Yes, the second they are two they’re supposed to have a ticket.
Whether or not anyone asks, who knows? Once my kids looked like they could talk, the TSA agent always asks them what their names are in order to match it up to their tickets, but no one’s ever asked for ID and I’ve never had an actual ID of any kind for them with me. (One year I had to explain VERY FIRMLY to my son that this was one instance where his three-year-old self telling everyone his name was “Baby Kitty Cat” was NOT going to happen.)
Instead of worrying about it, wouldn’t it be easier to just buy a ticket? (And also a lot more comfortable. I can’t imagine having to sit with a squirmy toddler on my lap for hours on end.)