The Allies weren't slurping up a lot Middle Eastern oil, because for the most part the Middle East hadn't yet become a major oil producer.

I'm pulling stats from International Historical Statistics by BR Mitchell.

Persia was by far the biggest Middle East oil producer during the war years. They generally produced a bit under 10 million tons of oil a year. 9.7 in 1939, 8.8 in 1940, 6.7 in 1941, 9.6 in 1942, 9.9 in 1943. Only in the last two years did they break that mark.

Iraq was generally at about 1/3 Persia's production. Kuwait didn't start production until after the war. Bahrain, Egypt, and Saudi Arabia all had minor operations that produced 1 million tons or less a year each.

In contrast the US was producing between 180 and 190 million tons per year before entering the war, and 220 to 230 million tons per year after entering the war. The USSR was producing at least 30 million tons per year in the first half of the war which slowly dropped to 19.4 million by 1945.

And that's just the biggest players. Canada was outproducing Brunei. Trinidad & Tobago was outproducing Egypt and Saudi Arabia. Mexico was outproducing Iraq. And Venezuela was outproducing the whole of the Middle East, for most of the war years.

The whole of the Middle East production was only around 7% of the US production alone. Add in all the other sources and the Allies were swimming in oil sources.The Middle East was a nice bit of extra oil, but it never accounted for more than a drop in the bucket... so to speak.

As to how they acquired the Middle East oil... they paid for it. The oil was being produced by western companies under agreements with the local governments. When they bought oil from Texaco in Saudi Arabia it was no different from buying oil from Texaco in Texas. Most of the oil agreements predates the war, for example AIOC (Anglo-Iranian Oil Company) operated under an agreement signed in 1935. Even when the Soviets and British occupied Persia in 1941, they continued to operate under the pre-existing agreement.