Contact the state agency that issued the 1099-G and ask them what their state ID is, or look on their web site. If neither of those ideas works, try using the federal tax ID, but with no hyphen.
It can't be that your 1099-G is the only one with the state ID missing. They probably sent them all out like that. There must be thousands of people in your state with the same problem. Maybe other people have already posted the missing ID, and you can find it on social media or with a web search.