Oil is useful for a lot more than just burning it. Fertilizers, plastics, and asphalt are the most well known but a host of industrial processes use it as a feedstock. It’s only recently that oil has been hard to find, initially it was seeping up from the ground which still occurs on the sea floor in some areas making it hard to miss.
So some other civilization could exist without using it, but they would look nothing like ours.
It is one thing to use it, entirely another to exhaust it. A civilization consuming only 1% as much could look very similar, if it were not dependent on oil for fuel.
It’s not just plastic but also Asphalt, Lubricants, Fertilizer, etc. However, let’s assume they only used 10% as much oil as us and only used oil for ~100 years before dying off.
They would still have started with easily extracted sources not offshore oil or ultra deep oil. The world had very easily extracted oil when we started. Further, there is no missing oil based on geology.
Most dry wells are around oil fields. We have fairly good understanding about the overall geology of oil fields, but you can have a dry well 50m from an active well. What we don’t have is most expected oil fields without oil.
So some other civilization could exist without using it, but they would look nothing like ours.