You will only need their EIN or tax ID if your daughters stay there while you went to work. If the gym provided child care as you worked out, it is not deductible and you will not need the EIN or tax ID. If your daughters stayed there while you went to a paying job, you can complete the return and if they still refuse, then you will need to send your provider a W-10 (Request for Taxpayer Information and Identification Number).
Get the form at https://www.irs.gov/pub/irs-pdf/fw10.pdf, fill out what you can and send it via certified mail after making 2 copies for yourself. If the provider does not provide the info within 10 business days, then you can still file your tax return without that information. However, you won't be able to e-file it. You'll have to mail it in. When you mail it, include a copy of the W-9 you sent to the provider, along with a copy of the proof that you sent it to them via certified mail.
From the IRS and federal tax perspective, there is no requirement for a daycare provider to be licensed. But if they provided you childcare services that you qualify to claim a credit for (if you don't qualify, then it's a moot point) they are required to provide you either their SSN or EIN so you can claim the credit you are entitled to. (Again, assuming you are entitled to claim the credit and can do so legally.)
I can understand why someone would not want to give you their SSN. You understand too. But maybe the caregiver isn’t aware of just how fast and easy it is for them to get an EIN (Employer Identification Number)? Make them aware that they can go online to http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Apply-for-an-Employer-Identification-Number-(EIN)-Online and it will only take them about 10 minutes to get an EIN – which they can then provide to you as required by federal law.
Assuming you have already requested this information from your dependent care provider, and they still refuse to provide it. here is the procedure that you must follow, in order to claim the dependent care credit without the provider’s SSN or EIN.
First, understand that you can not e-file your return without the caregiver’s SSN or EIN. You will have to print, sign and mail it to the IRS. There is no alternative or workaround for this.
Next, print out a form W-10 (Request for Taxpayer Identification Number and Certification) from the IRS website at https://www.irs.gov/pub/irs-pdf/fw10.pdf
Fill out the W-10 form with all the information that you know.
Make 2 additional copies of the filled out W-10 for your own records.
Mail the form to the caregiver via certified mail. Don’t lose your post office receipt.
The caregiver has 10 calendar days to respond. If they respond with the information you need, then you’re good to go. If they have not responded, then on the 11th day after you mailed it certified mail:
Select the “print and file” option for your tax return. This will print only the documents required for filing, along with an instruction sheet that contains the address of the specific IRS processing center you will mail it to.
Include in your mailing a copy (not an original) of your proof of notification (the certified mail receipt you got at the post office), and a copy of the W-10 that you sent to the caregiver.
That’s it. While this will delay your refund (it’ll be 6-8 weeks before you get it), you’ll still get the credit. At this point, it’s no longer your problem – it’s the caregiver’s.