Vermillion is absolutely "Big Oil." They are an oil and gas company that make several billion a year. If you don't consider that "big" I don't know what is.
Regardless, Vermillion is the only oil company that sued anyone and did it a few years ago. Where’s the evidence of Big Oil currently suing?
Companies like BP, Total, etc operate more like nation-states and have other tools available.
I'd still call them "Big Energy" however.
Exxon: $440B