I think this is going to depend upon what exactly you mean by an "existential question".

Typically, Kierkegaard is the earliest person taught in your standard existential philosophy seminar. However, questions that could be deemed existential are far older; people have likely been worrying about them since time immemorial.

For example, Aristotle spends quite a bit of time in the Nicomachean Ethics discussing what it means for one to live the good life, concluding that the good life consists of the attainment of "happiness" via the fulfillment of one's function as a rational being. One could argue here that Aristotle is here answering the existential question "What is the purpose of life".

Other questions that might plausibly be seen as existential in character are "What does it mean to exist as a human individual?" and "How should the individual relate to their mortality?"; these sorts of questions pop up nearly incessantly in the history of philosophy and world literature.

Basically, I think you will need to refine your question a bit to facilitate getting the best answers you can.