The USD is not specifically dependent on oil prices. That is just one in very many things that influence the value of the dollar. The bigger ones are interest rates and trade flows.
The reason the oil producers "accept dollars" is simply that global oil prices are denominated in them. I'm sure there are transactions done in other currencies, but it all ties back to the market price for oil, which means a USD relationship.
I have no idea what the history of that is, though. Probably a post-World War II thing when oil production was really ramped up globally. In generally terms, that's probably when the USD became the dominant one in the world. Before that it had been the GBP, I believe.
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