When you leave employment from a job, you are permitted to withdraw your 401(k) (take a distribution), or rollover or transfer the money to a different qualified plan like a private IRA or retirement plan at your new employer. You don't have to certify for COVID. If you are under age 55, and you take a distribution, you pay income tax plus a 10% penalty on your tax return, and the plan administrator is required to withhold 20% for taxes. The 20% is not the total tax bill, it is a backup amount, and the actual tax and penalty is calculated on your tax return.
If you can certify that your distribution is due to a COVID hardship, then you don't have to pay the 10% penalty on your tax return, and you don't have the mandatory 20% backup withholding. The plan's "COVID form" is probably used to exempt you from the mandatory 20% withholding.
Note that this does not change your ultimate tax owed, just when you pay the tax. If you can certify on your tax return that your withdrawal was due to a COVID hardship, you can spread the income out over 3 years, or you can take all the income at once and pay the tax all at once. If we assume that you withdrew the maximum of $100,000, and spread it out over 3 years, your income tax will still probably be $8000 higher than last year, and you will need to come up with that money out of pocket if you don't have the mandatory withholding.
