Their competitors - which like the US military will be state-controlled entities - for burning the most fossil fuels aren't exactly known for their transparency. They're generally going to be in China.

And then to be able to burn any, you first need to take them out of the ground. If we count those indirect emissions, obviously Saudi Aramco causes orders of magnitudes more emissions than the Pentagon. Sure, this comparison may be "cheating", but put it this way; it's much worse for global emissions to have Aramco improve their barrels/day rate by 10% than the Pentagon's budget increase by 10%.