Once you near the objective it should pop up a menu option box that asks if you want to launch the raft or postpone. If you don't see that check in the upper right hand corner of the screen. If there's a symbol there click on it and it should pop up the menu option box.
Make sure you're stoped or just barely move'n along. My sub was move'n around 3 kts and hit the raft damaging it so it wouldn't move. I couldn't complete the mission and had to reload my last save game. I also recommend that you don't use time excell above 8x as sometimes the raft will flip upside down especially if it's in rough waters. Another minor game bug.
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