What are the odds of my wife securing a position at Aramco?

Has she applied for Aramco jobs yet? If she see jobs that are a match, then it’s possible but no guarantee. There are dual working couples at Aramco. Not a lot but some.

Not sure, if your recommendation would carry any weight since you have not started working here yet and you’re her husband. She should apply for jobs at Aramco.jobs as most do. She can apply now and continue applying after she arrives for jobs that she believes she is compeive for. She may mention in cover letter that you’ve been hired and she’s moving to Dhahran. This may give her a little edge, if the hiring dept reads it.