I had a son 21 years ago in Kansas and didn't tell the father. I lost contact with him before I knew I was pregnant. I have tried on several occasions to find him on my own without success. If I am able to find him, can I make him pay back support, since I've been attempting to find him? And who would get the money, me or my army soldier son?

Laws on this go by state and I don't know of any state that allows you to file for the first time after the child turns 18.
Plus you would have to get the father to either agree to DNA testing, or you would have to get a court order for it.
If you had been awarded back support, the money would go to you.

Child support is paid to the custodial parent. But I doubt if the courts will allow you to file. At this late date.

You COULD have file in absentia after the child was born. Had you done so, if and when he was found, you would have a standing order to enforce. Now, I'm afraid that boat has sailed.