You have two options:
1. Request an extension and keep trying to get a copy of your 1099-G - as @SweetieJean pointed out, many states only provide the 1099-G online. "Regular unemployment" and "unemployment compensation " are the same thing, so if your state has an "Unemployment" website, you should be able to get a copy.
2. File now, without the 1099-G. It is likely that your unemployment compensation is not taxable, so a "missing" 1099-G probably won't affect your federal tax calculation anyhow. See New Exclusion of up to $10,200 of Unemployment Compensation.
Option 2 has the benefit of speeding up any refund you may have coming. If your unemployment qualifies for the recent tax-exemption, then the "missing" 1099-G won't change anything on your federal tax, anyhow.
If your state is still taxing unemployment income, they'll take note of the missing 1099-G and send you an adjusted tax bill later in the year.
They will calculate the tax due (plus interest and penalties) and you can send them a check.
The IRS and the states use this simplified procedure since It is extremely common to omit reporting a Form 1099. You file before they arrive, they get lost in the mail, get lost in your house or get tossed along with junk mail.
