It's about money. If you had to pay a mortgage each month on a vacant house and the alternatives were to spend money to fix it up to sell or lease it to someone right now so you could pay the mortgage, which choice would you make?
You're paying their mortgage. Would they sell? Probably. As an incumbent you have the opportunity to see the king without his clothes. You're living there and know its faults. You could ask a Realtor what it's worth and put together an offer to purchase. There is less pressure on them to sell at a reasonable price now, but the opportunity for someone else to be on the hook for the mortgage payments would be tasty. And, the risk that you might move out and buy somewhere else at the end of your term may be enough incentive for them to make a deal.
You could start by mentioning to the landlord (not that you want to buy his place) but that your long term goal is to find a house and buy it. If he wants to sell, you'll get the suggestion from him almost immediately. To protect yourself, you still want to do an inspection and you would probably be served by having a Realtor put together the offer for you. Make sure you're able to get a loan to buy it first.
Look, you know at one point they were interested in selling. They probably still are. They rented to you, probably, because they couldn't find a buyer. You can't know for sure if they're still interested in selling, but it is a reasonable assumption.
You ought to do some homework first. Find out, approximately, what the home would sell for in today's market. A Realtor can run a CMA for you. And, no, you don't have to work with that agent in the purchase of the home, although it ultimately might be a good idea. So let's say a CMA comes back saying your home is worth $400,000.
Then there are two ways of approaching it. One is a formal offer; the other is an informal approach. I'd suggest starting off informally. Contact the owners and explain that you really enjoy the house. In fact, you enjoy it so much you might be interested in buying it. You remember that they were interested in selling it, and were wondering if they still are. If so, you'd really be interested in trying to work something out with them.
They may give you an indication of how much they want for the home. Or not.
I don't know the customs or the laws in Texas, but it might make financial sense to have a lawyer lined up to do all the work for you. A Realtor would be more efficient, but--and here's some heresy--you may not need one in this case. You do need professional assistance in the transaction, but if there's a "meeting of the minds" between you and the owners, then it's largely a matter of committing the agreement to paper.
One thing to be careful of is to keep the lease and the purchase separate. In most areas of the country, you might transition into the purchase via a lease-purchase but my understanding is Texas is very, very restrictive in such matters. So be careful.
But I'd suggest you consider those steps: Determine the value of the property, line up someone to help you with the transaction if it works out, then make an initial informal contact with the owner. (Others here may advise getting the CMA only after the initial contact; nothing wrong with that either. And some may argue in favor of using a buyer's agent, not just a lawyer. That's OK, too.)