I have searched Google and there are more than 200.000 cases using "at stake with". But, I'm dubious about the phrase "at stake with", here I want to say that the economy is depended gravely on the oil revenues.
"At stake" would imply that their revenues are in danger of failing in the near future…
The country's economy is at stake if falling oil revenues continue
You perhaps want "staked in" which still implies some kind of gamble..
The country has staked all its investments in oil production
If you simply want to say they are highly dependent on oil, with no immediate danger of that revenue going away any time soon, then
The country is highly dependent on its oil revenues
would feel a much calmer statement, though still acknowledging that if in the unlikely event of those revenues failing, they would still be in some trouble.
Is there anything wrong with saying your meaning? That is: "The country's economy depends gravely on its oil revenues"? Gravely is an odd word choice but the sentence makes perfect sense. A more common word for the sentiment you're trying to get across would probably be "heavily".
Using "at stake with" implies an immediacy to the statement and also leads to questions to fully understand the meaning, which requires information not present in your original statement.
To explain that, "at stake" can mean "at risk" or "could be lost", so if you substitute that meaning into your opening statement: "The country's economy is at risk." someone listening would want to know "Why is the economy at risk?". However, "Its oil revenues" doesn't fully answer that question and just leads to another question: "What about the oil revenues?". You could say "The oil revenues are dropping." or that they're too low and that would complete the statement. Saying "The country's economy is at stake, with its oil revenues dropping" works, though it feels a bit clunky to American English. Inverting it to say "With its oil revenues dropping, the country's economy is at stake." flows much more naturally from cause to effect.
You could say "the economy is staked on oil revenues" or "The country has staked the economy on its oil revenues", which implies that the country has wagered its economy on oil and now depends on it. Using "staked" also implies the economy would be lost if oil revenues were ever lost.
However, it's the wrong statement to make if you just want to say the economy depends on oil revenues as the original statement that the economy depends on oil revenues has a much weaker link between the two and implies far less dire consequences.
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".
First, I agree with you. This is an awkward construction--one that seems as if it ought to work, but doesn't. You need to have either an agent that's placing the economy at stake (use "due to" or "from" and indicate what has changed to place the economy at stake) or you need to switch from "is at stake" to "has been staked". Examples: "This country's economy is at stake due to (rising? falling?) oil revenues" or "This country's economy has been staked on oil revenues.”
However, I'm seeing another issue: the use of the word "revenues." If you're talking about the U.S., I suspect you mean oil prices. Revenue refers to income. Oil companies receive revenues from the sale of oil. The U.S. as a whole does not. Example of one way to change this: "This country's economy has been placed at stake by rising oil prices."
If you're referring to another country, one with a government that directly receives income from the sale of oil, you still need to say what's happening to the revenue to place the economy at stake: "This country's economy is at stake due to the steep decline of oil revenue" or "The steep decline in oil revenue places the entire economy at stake."