Which, can be tricky by the way.
Basically the president is the Chief of the armies. He is the highest commander in charge.
For rescue operation in a foreign territory, he basically cannot not be aware and not authorise the operation.
For a smaller operation, as Artemis, the president is authorising but the parliament can/must be involved.
For a war, the parliament his the one giving the go ahead after proposition of the government. The Senate also has a voice.

Well, that's the general frame. After, you have the secret operations which can be decided by others than the president but never in contradiction with his orders.

There is theorically very little room for a vaccum in the chain of command. Now, some orders can be interpretations...