I doubt very much you would be able to line up summer work in the Oil & Gas sector over due to visa reasons as well. If you are working for an operator over there and they sent you here it's a different matter. Just rocking up and getting a summer job is pretty tricky.
Abu Dhabi is where it'd be and it's unlikely they'd give you a chance here, they take alot of locals from university here and have quotas.
Also, most recruitment companies wouldnt do much work with fresh graduates either tbh, we get paid for proven quality and not potential, plus paid on fees and there isnt much money in graduates from our point of view.
That said, i would definitely recommend contacting them yourself and not relying on agencies.
Do you specifically want to work for an operator ? They pay more but you might want to consider a service company like Schlumberger or Weatherford, who give excellent training. A few years with a company like that is a better way into a career with an operator
imo
'Shell' take alot of graduates and they do alot of recruiting themselves. There are several oil fairs around these days where operators have representatives you can speak to. I think there is one coming up in Kazakhstan soon, which may be a little bit of a trek for you
I'd try Shell, Occidental and maybe someone like Anadarko and definitely Weatherford & Schlumberger. Service companies are always crying out for Petroleum Engineers with experience and Schlumberger like to train a specific way.
Also, 'Austcorp' are a pretty big recruitment company in Oil & Gas out there. They may have contacts within Aussie companies. If you're out there it's where you're more likely to get something tbh