Vergil (that is a more accurate spelling of the name) was part of the Augustan age. So, his influences must have been those religions and philosophies of his day. So, the question is what was around?
First, we should probably say that Vergil got a lot of his material from the epics of the past, namely, Homer. The very first words of his epic show this. I sing of arms and a man. The arms = Iliad, Trojan War. Man = Odysseus. So, the religion of the Greeks probably influenced him.
Second, Roman religion influenced him. Roman religion was very much into religious rites. In short, Roman religion was about "right doing" and not so much about "right believing."
Third, there was a lot of philosophies. Epicureanism and Stoicism come to mind.
I will try to answer your question. I am not really an expert in ancient theology but from my reading of The Aeneid, I sense a similarity between Virgil's view of his theology and that of Aristotle or Plato. Both philosophers were skeptical when it came to their polytheistic religion. The way Virgil depicts the gods in his epic is quite reminiscent of what Plato said about the gods: ravenous and too human to be gods. According to Voltaire, Aristotle was also accused of atheism. For instance, consider the first book of The Aeneid. Already the poet questions the insatiate ire of Juno and how could divine breasts entertain such an avengeful passion. Considering the immense piety of the Romans, this peculiar depiction of the gods as versatile and wavering is very interesting. If there is an influence that may have brought about such a vision of the deities, then to me, it is definetly Greek. I am aware that I have said only a little bit but that's unfortunately all I can say about this.