I do not believe SQL Developer would have DataPump utility.
As the name implies, SQL Developer is intended to be used as a tool to facilitate development (typically of SQL code), as well as provide additional features to assist developers in their daily routines.
DataPump, is more of a DBA tool used for extracting logical copies of data from an existing database and/or port to another database. To use this utility, you would typically need to be in the dba group (although not necessary if configured properly). But at least on the same server as the datapump utility software resides.
There are many options and features to using datapump, including network_link which allows extraction of data from a remote database to another database on a different server without ever creating a file on the operating system.
If I recall correctly, I believe in older versions of Oracle client (e.g., 8.x), the Oracle client provided the export/import utility. But those days are long gone now. With datapump, it is primary a DBA tool.
Data Pump server code is always installed and I thought that the Data Pump client was installed by default. At your operating system prompt just type:
If you get a response, then the Data Pump client is installed. Data Pump server code can be called by the client and/or by the API.
Hope this helps.