Is it permissible for me to submit personal photos without a headscarf to a military academy that requires me to do so to complete my employment application, given that I am allowed to wear the headscarf during work?

A woman's head is considered awrah (private, to be covered) by scholarly consensus. It is not permissible to uncover it in pictures except out of necessity or a dire need that is treated as a necessity, such as the need to work to support oneself when no alternative is available.