The form is asking for a US zip code for my child care provider who lives with me in the United Kingdom.

is the provider a US citizen/ resident -- then use SSN and ignore about  zip code -- foreign address with foreign post code.

If  he/she is local resident/ citizen, then ignore the details --- the  requirement for the details  is because IRS wants to be able to  see that your expenses are  reported as income on another US taxpayer ( if not  US, it does not mean anything ).