If you are not an employee of the daycare provider, they will not issue you a W-2. If you are an employee, then by federal law they are required to issue the W-2 not later than Jan 31st. Period. There are no exceptions for that.
If you are a client of the daycare and have not paid fees due for 2019, paying them now will NOT help your 2019 tax situation. They are required by law to provide you a statement that provides their EIN along with the amount you actually paid them in 2019. If they don't, simply send them via certified mail, return receipt required (not requested) an IRS Form W-10 which you can get at https://www.irs.gov/pub/irs-pdf/fw10.pdf
If you send it after Jan 31, they have 10 days to respond. If no response then you can file without it. But you won't be able to e-file. You'll have to print, sign and mail the return to the IRS along with proof that you made a concerted effort to get the EIN.
The debt you may owe the daycare is not a tax issue. It's a legal issue and if they so choose they can refuse to provide you future services with no notice, as well as take legal action in small claims court for the debt you may owe.
