This information is in the upper left corner on all actual Forms 1099-G. If you do not see the information on your document you should contact your state unemployment agency for the information or location and instructions for online access to print the document.
This could eliminate e-file eligibility without it. If you have difficulty obtaining the information you can enter enough information to file a paper return. If there is any federal withholding on your document it will be required to be sent with your paper file as well as any W2s.
Thats odd. Do you have an actual 1099-G? Or some letter that is "in lieu of" the actual form? See the actual form at https://www.irs.gov/pub/irs-pdf/f1099g.pdf As you can see, the top left box is where the payer's name and address is entered.