While Karma outranks Irelia in terms of politics and the social welfare of Ionia, who outranks who on the Fields of Justice? Who was giving the orders during the Invasion of Noxus? Basically, who's at the highest chain of command when ever there is a conflict?

Irelia, the Ionian Captain of the Guard

or

Karma, the Duchess of Ionia

karma, cuz more ppl play her so she wins more

Neither, actually. Irelia's brother has supreme command over Ionia. Irelia is captain of the guards, but still takes orders from her brother, who took their father's place when he died.

We actually don't know what happened to Irelia's brother. The last we heard about Zelos was that he traveled to Demacia to get help. We never heard back, so he could be lost at sea for all we know.

As far as is known, Ionia is ruled by multipal clans. It's not like Noxus where there's a single ruling party.

I'm sure they do. But in a conflict, especially in war, there needs to be a least one person that stands up and says "follow me".

While the Ionian Elders may decide as a group the best strategy and direction to take, there needs to be someone with the final say on the best tactic fulfill that strategy. To have multiple people at the very top of command would just lead to slow decisions, possible arguments, and miscommunication.

In politics, group thinking can work by itself with no central leader. In war, due to the need for quick decisions, a central leader is needed, at least on the battlefield.

I assume from her Judgement that she was given the political title of Duchess of Ionia in recognition for her deeds on the battlefield during the Noxian invasion.

Irelia seems to hold some military rank, but Captain of the Guard, doesn't seem like it is a top-tier position.

I would say Karma out ranks Irelia in the military structure of Ionia.

However, either the Ionian Elders, Irelia's brother, or another yet unnamed person(s) would be who Karma would get her orders from during a military conflict.

Well, in her lore, Karma was "made the figurehead of the Ionian resistance". While I'm sure the Elders collectively made the decisions, Karma seemed like the one giving out the final orders.