As others stated, you need to consult a lawyer who is familiar in this area of the law.
That said, your biggest issue here is the "verbal" contract. Being all related further complicated matters. Many authorities will view this as a "family dispute."
In most states, a barn can put a "mechanics lien" on a horse and/or tack for non-payment of board. Some area may have time constraints on this (after 30 days, after 90 days, etc.). This can usually also be specified in a written contract.
From a legal standpoint, a horse (typically) is not a "pet" - it is livestock. That means it is property with a value, just like the tack that was left behind.
Board has a value, other care provided has a value also. If the agreed upon board is $500/month and you haven't been paid in three months, then you are owed $1500. If you sold the horse for $2000, you would keep $1500 and pay $500 to the old owner. That is usually how it works. If you plan to keep the horse, it is more complicated since an accurate value of the livestock has not been determined.
Getting to that point, however, is a bit more complicated.
With only a verbal agreement, the old owner can claim just about anything. "They agreed to six months free board." "I paid them in cash." ...or a million other excuses/lies that becomes a "he said/she said" in court.
First thing you will need to do is notify the person in writing of the situation. State clearly what is owed and for what time period. Give a reasonable time period for repayment (reasonable may vary by locality and situation) and clearly spell out the consequences for not repaying. "Full payment is due by December 10 or the horse will be sold at auction and the proceeds used to satisfy the outstanding amount." Something along those lines - contact a lawyer for the best wording. Notice should also be sent by a "certified/return receipt" type of mail so you can prove they received it.
You really need to have your i's dotted and t's crossed to go ahead with something like this. Especially if you plan on keeping the horse involved. The old owner can always return to get "their" horse and ignore our claim of ownership. Then they sue you!
Small claims court is a good idea. If you win, then you have an actual court judgement in your favor for a specific amount. From that point, it would be much easier to take the horse/tack as payment for that judgement.
Again, seek out a lawyer for some real advice on this.