Is this package missing and if so, how do I fix?
Data Pump is a fully integrated feature of Oracle Database 11g. Data
  Pump is installed automatically during database creation and database
  upgrade...more


It should be installed during the database creation. I found logical error in your query to check whether or not the data pump package is installed. The object type PACKAGE should be in upper case to be selected. 

Demo:

You query has returned the following only.

SQL&gt; SELECT object_name FROM ALL_OBJECTS WHERE OBJECT_TYPE IN ('FUNCTION','PROCEDURE','package') and object_name like '%DATA%' order by OBJECT_NAME;

OBJECT_NAME
------------------------------
DATABASE_NAME
DBMS_FEATURE_DATABASE_VAULT
DBMS_FEATURE_DATA_GUARD
DBMS_FEATURE_DATA_REDACTION
DBMS_FEATURE_EXADATA
EXF$VALIDATEDATAITEM
ODCIANYDATADUMP
XDB_DATASTORE_PROC

8 rows selected.


And the query with object type PACKAGE in upper case has returned the following. You can see the DATAPUMP package is there.

SQL&gt; SELECT object_name FROM ALL_OBJECTS WHERE OBJECT_TYPE IN ('FUNCTION','PROCEDURE','PACKAGE') and object_name like '%DATA%' order by OBJECT_NAME;

OBJECT_NAME
------------------------------
...
CWM1_OLAP_VALIDATE_METADATA
CWM2_OLAP_METADATA_REFRESH
CWM2_OLAP_VALIDATE_METADATA
DATABASE_NAME
DBMS_DATAPUMP
DBMS_DATAPUMP_UTL
DBMS_DATA_MINING
DBMS_DATA_MINING_INTERNAL
...