The Gulf Monarchies would have sold them oil in the same way they have been doing to the US as hegemon.

And the USA has significantly intervened in Middle East politics.

It is likely that Nazi Germany would intervene in the Middle East to secure its oil; the various nation-states were still forming and borders were somewhat volatile. Germany would likely intervene to ensure that the oil flows.

Also, after a while, the German state would need another infusion of money and cheap Arab oil would be a perfect resource to bring under control of the Nazi state.