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.
