May 17, to pay
Their failure to provide a 1099 does not give you an extension to pay.
However, if you don't pay on time, filing a Form 4868 extension will remove some penalties. I recommend paying on time.
April 15 2024, to file, for all practical purposes
Their failure to provide a 1099 does not give you an extension to file.
However, there is no penalty for filing late if you are owed a refund.
You have 3 years to file if you want your refund.
Your last paycheck stub should have all the info that your 1099 would give, since it includes running totals of the relevant numbers so far. If you are paid by electronic deposit, the stub still exists. It would have either been postal mailed, or accessible as a downloadable PDF from the website of the agency through which they pay. A contact to HR will either get you the 1099 or last pay stub, or the name of the third party company who does have that. Many companies use third party paycheck issuing services, such as Paychex.
If it is simply impossible for you to determine from available data, then you need to estimate your taxes owed based on the data you do have.
If this will have you anywhere near owing tax, I strongly advise you make a payment by filing a 2020 Form 4868, essentially using that as a payment coupon to assure the IRS credits the right taxpayer's account. Overpay, so that you are owed a refund. Now, there is no penalty for filing late.
"I don't pay with paper checks, I e-file" — that'd be a problem. E-filing isn't really filing, it's getting the IRS to agree to data interchange in lieu of filing, and IRS doesn't need to agree to accept that. If your data is incomplete, they probably won't agree to accept that... so you need another option.
Paper filing (if only the 4868 + estimated payment) is something the IRS cannot and will not refuse.
They will not refuse a paper filing over a quarrel about the amounts. That gets resolved other ways.
The point is, get your money in and your toe across the "payment" finish line, and that gives you time to sort out the rest.
(E?) Payment is indpendent from (E?) Filing, so if you owe money and can in some manner e-pay to make that estimated tax payment, then by all means do so.
P.S. if you didn't get your 1099 because of ever-varying postal addresses because you move a lot... seriously consider a PMB.