Some random thoughts:
The LO is responsible for paying this back, legally and morally.
Pretty much zero chance he pays everybody back. Getting paid would require that he feels the moral responsibility to pay it back, is actually willing to take this amount of loss, and that he has the means to be able to pay it back. There is only a small chance for any one of those things, and almost no chance for all of them.
Most homeowner's or renter's policies, assuming he has one, have exclusions for amounts of cash they cover, have exclusions for covering business related things (even if the business is conducted from the home), and have exclusions for covering other people's property (and this cash belongs to other people, not to him). I would insist that he provides proof of having filed an insurance claim against whatever insurance he has just to be safe, but I think the realistic chances that it covers anything are almost non-existent.
Pretty much no chance that the national league organization pays it. It isn't their responsibility, legally or morally. And it is too much money to do it out of the goodness of their heart. And even if it wasn't, they would be afraid it would incentivize future LO's to "lose" or have their league money "stolen" knowing that the national organization would just take care of it. Also, if they covered this type situation for one guy, but not for the next one, then it looks like they are playing favorites, etc, so any way you cut it I don't see the national organization doing anything.
If you go to him and ask him for your cut, even if he has the ability to pay it, you are likely to be met with "it isn't fair for me to pay you when I can't pay everybody else". Still worth asking though if the thought of getting paid when others aren't going to be paid doesn't bother you. If you are going to do this then do it immediately. The chances for the first guy or two getting paid are infinitely better than for the 15th guy that asks.
If you think you would take him to small claims court over this (there is usually about a $150 fee to file this type suit and you both represent yourselves in court--sounds like you would have a strong case assuming you take proof of your league participation/payments and proof of whatever prize you won), then you want to do it sooner rather than later. The first guy or two with judgments are a lot more likely to get paid than anybody down the line because even if he is willing to pay the first or second person that took him to court (probably only the first guy at best), at a certain point he is going to say screw this and stop paying, judgments or not. If he was willing and able to pay $19,000 then he would have done it to begin and the same unwillingness or inability to pay then is going to make him stop paying judgments at some point too.