To join a fleet in Star Trek Online, you need to be invited by a fleet member that has invite permission. You can only receive a fleet invite if you are not currently in a fleet.
Follow these steps to join Stargate-Union.

First, click the "Social" botton, which is to the right of the mini-map.

Next, click the "Find Fleet" tab.  In this tab, you will see a list of fleets.
Click the dropdown arrow at the left to open the search box.

Type Stargate-Union and then click the search button.  Select the fleet name in the fleet list, then click the "Details" button.

The search system is buggy at times, so if nothing shows up, wait a minute and try again.

The fleet details window will show a list of fleet members that are currently online and have invite permission.

You can send a private message to any member asking for a fleet invite.

If there are no members currently online, you can send a mail to @blevok or post on our forum and we will send you an invite when we log in.