Changing SQL Server installation - Will it bring the server down? I need to install SSIS on our server, one concern that I have is the database being brought down temporarily since it's mission critical that the server is always up. 

Does changing the SQL Server installation disable the server?

If it does bring the database down, is there any means of installing SSIS without touching our main installation? I could always create a fresh install if this is more suitable.
Adding SSIS to your server should not cause the running SQL Server to be affected, stopped, restarted, etc. I have never seen this happen with adding SSIS to a server.

However, if you want to make absolutely sure, you could test by installing SQL Server on a test server.  Then come back and install SSIS and see what happens.