You need to escape the other {
and }
using {{
and }}
:
>>> print 'https://app.example.com/SearchService/search/xref?parts=[{{"partNumber":"{0}","manufacturer":"my company"}}]&fmt=json'.format('my product')
https://app.example.com/SearchService/search/xref?parts=[{"partNumber":"my product","manufacturer":"my company"}]&fmt=json
From docs:
If you need to include a brace character in the literal text, it can be escaped by doubling: {{ and }}.