If the underlying database fields are text instead of number then you need single quotes too:
RsCUSTOMER.open "Select Customer_Name from CUSTOMER WHERE Customer_Surname = 'the value in the text box' AND Customer_PostCode = 'the value in the text box'", Conn
And with actual variables might look like:
RsCUSTOMER.open "Select Customer_Name from CUSTOMER WHERE Customer_Surname = '" & TheTextboxVariable & "' AND Customer_PostCode = '" & TheOtherTextboxVariable & "'", Conn
And of course hope there aren't any single quotes in the last name or postal code or else you'd want to double those up which is probably a good idea anyways as you never know:
RsCUSTOMER.open "Select Customer_Name from CUSTOMER WHERE Customer_Surname = '" & Replace(TheTextboxVariable, "'", "''") & "' AND Customer_PostCode = '" & Replace(TheOtherTextboxVariable, "'", "''") & "'", Conn