Yes, Address Finder Web Service supports partial addresses. If you do not provide the complete address, you can still get a list of candidates based on the information you have entered. For example, if you only enter a city name, the result list contains cities whose name matches the name you entered.
v2006 and v2006.1 - Request that Address Finder returns partial addresses; by default, this functionality is turned off. To turn on partial address support, first use the method getInfo to see which data sources support partial addresses, then set the object parameter AddressFinderOptions.partialAddress="true".
v2.0 - All data sources with the word 'Address' in the middle of the data source name, such as TA.Address.US, return candidates with partial address information.
Using a) the name of a street, b) the state or c) the street with the state returns Error 6001: Not enough address information.