Yes, the PK4 TimeTracker works on Salesforce Professional Edition. However, you need to ask your Salesforce Account Executive to activate API access for your org, 

For more details, please refer to this article in the Salesforce documentation. Refer to the API Notes section that talks about the AppExchange apps. PK4 TimeTracker has the appropriate "Client ID" issued per our security review. API requests identified with the PK4 TimeTracker's API token are allowed to operate with Professional Edition organizations.

Note: Please understand that the PK4 TimeTracker API token only allows access to the APIs that the TimeTracker uses within the product. You will NOT be able to use any other APIs.