If I get prompted - "Form 2441 - Provider SSN must be entered." but, I don't have an SSN as the provider refused to share their SSN. So how can I proceed with this?

[xmasbaby0]

When a childcare provider will not give you a SSN or Tax ID

If they still refuse to provide an EIN or SSN, send your provider a W-10 (Request for Taxpayer Information and Identification Number).  You can get a copy of the W-10 by clicking here: W-10 Form & Instructions  Complete as much of the form as you can and send it via certified mail (after making copies for yourself).

If the provider does not respond within 10 business days, you can still file your return.  However, you won't be able to e-file.  Print and mail in your return, along with an explanation and a copy of the W-10 you sent your provider.

https://www.irs.gov/individuals/international-taxpayers/taxpayer-identification-numbers-tin