I've certainly heard of people buying a house under contract in my area. It can be a good thing under the right conditions. The seller can only sell "under contract" if they own the house free and clear. Some sellers like this as it can be an extra income (sometimes the interest rate is more favorable to them than they could get in investing).

As a buyer, you need to get advice from a real estate attorney and make sure that documents are drawn up or reviewed by the attorney. I think that the down payment would be negotiable. There are some pitfalls -- a young acquaintance is in a problematic situation. Her father unexpectedly died (age 58), her mother is in intensive care. The father had been paying for the house "on contract" for a number of years. She hasn't located the contract papers. The owner of the house is harassing her, saying that she wants to inspect the home.
The legal aid attorney is saying that the young woman is not authorized to allow entry to the owner (the hospitalized mother is now the only person with the authority). It's unlikely that the mother & daughter can afford the monthly payments -- so most likely the many years of monthly installments on the contract will be gone.

Good luck to you.
Susan