What you want for this, assuming you have intelligent testers who aren't expected to follow a "click this button" script, is to define one test case for customer lookup, and in that test case, list the ways to search for a customer.
Depending on how your system works, you should be able to define the search methods as test parameters that can be used in customer search - especially if you keep your description at a high level.
"Given that Customer X has a %PARAMETER% stored in the system, when I search for customers by %PARAMETER%, I should see the Customer X in the list of matching customers."
Parameters could be:
- Full name
- Contact name
- Zip code
- Birth date
- date of last purchase
Pretty much any of the fields that you could search on.
Note that you should also include test cases for customers who have nothing stored in the field e.g. customers with no birth date stored should never be returned when searching by date of birth; and for customers who have a different value than the search value e.g a customer who does not have zip code 12345 should not appear in the search results for zip code 12345.