If a women wear veil during studies but if a task is assigned to her i.e: to upload audio lecture on social media then Is it permissible that she can deliver audio lecture on social media. and If it is not permissible then what can she do, if the task assigned is mandatory.
Giving audio lectures is not per se haram for women.
Many scholars consider the voice of a woman as 'awrah (based on verses such as 33:32), but none of them has ever rejected ahadith transmitted by female sahabah (including the mothers of belivers, for example 'Aisha used to give hadith lessons behind a screen) or female scholars (the wife of ibn Hajar al-'Asqalani was herself a hadith scholar who got a certification from al-Hafidh al-'Iraqi one of the teachers of ibn Hajar himself and she used to teach while he was alive), which shows that it is possible to teach others (even men) for women.
So basically giving an audio lesson or lecture is halal for women if they respect the sharia rules on mahrams, 'awrah etc.. For example some scholars add as a condition that the women should speak in a neutral voice to avoid fitnah which is derived from the linked verse.
Yes, you can deliver your audio message without showing your face. There are lot of examples when a women came to S.A.W to ask any problem and knowing somethings from Prophet. And After the death of S.A.W the wives of S.A.W also deliver her knowledge as audio without showing her.
Indeed you can, Islam isn't that strict about women. You are free to do live audio, it is completely fine.