Can they sue me even though i've been making payments on the account?
The company can sue you any time within the statute of limitations.
I do not have a contract with them.
It certainly sounds as though you had a contract. You refer to "your" oil company, presumably were delivered oil, were billed for the oil you were delivered, and had agreed you owed the money for it. I don't see that happening if you didn't have a contract. If you meant you didn't have a written contract, then understand that most oral contracts are enforceable so long as the party suing can prove the contract. Under the circumstances here, it probably wouldn't be hard for the company to do.