It's not the only interpretation, but I read "This country's economy is at stake with its oil revenues" to mean, "This country's oil revenues are at stake, and the economy is at stake along with them".
That is to say, there is some risk/peril to the oil revenues, allegedly the risk was entered into through a specific decision in hope of gain (since otherwise it's not "at stake", the gambling analogy doesn't apply, it's merely "under threat"). Also, the economy as a whole will be won or lost along with the oil revenues.
If the oil revenues aren't "at stake", merely "under threat", then I'd say it's better not to say something is "at stake with them". Granted, this is a pedantic point, because the difference between "being at stake" and "being threatened" is often negligible. But if you want to be really precise in what you're saying, then you'd have to account for it.
If you mean to say that the economy depends on the oil revenues, and that allowing that dependency to exist is a knowing risk, then you could say "The county's economy is staked on its oil revenues".