Random = that which is proceeding, made, or occuring without definite aim, reason, or pattern.
So if it lacks aim, or if it lacks a reason, or if it lacks a pattern, it's random.
"Aim" kind of makes it redundant. Since aim implies a purpose, which requires an intelligence, then of course it would lack aim without there being a God. You might as well ask, "Are there any options for the universe coming to existence other than God or not God?".
"Reason" can easily imply the same thing as "aim". Of course it could be said that the universe came into existence
because of, say, a quantum fluctuation. So there it has a reason. Unless you mean "reason" like you mean "aim", and that is to imply purpose. So again it would be, yes, the universe either exists because God created it or not because God created it.
Pattern though, is a bit different. If there is only one universe, then there can't be a pattern. It isn't really a valid question to ask of it. Just like we wouldn't say the existence of God doesn't follow a pattern. It is singular. If there is a multiverse, then it would be difficult to determine if there is some overlying pattern in the way that universes pop into existence. Maybe there is, maybe there isn't.
I think "random chance" might be a bad way to phrase it. It doesn't apply well to offer a distinction between created and naturally occurring.