SQL Data Collector - SSIS required on source server?
Nope, you don't need SSIS on the server you are collecting data from, bits of SSIS are installed anyway when you install SQL server and this is enough for the data collector to work.  All the system data collector sets work out of the box once you have set up the management data warehouse and enabled them.

The actual SSIS packages for data collector are stored within MSDB, rather than SSISDB as with a full SSIS server and they are executed by a process called Dcexec.exe.

The full architecture for how data collector works is detailed here:

Data Collector Architecture