Did you claim a C-Store/Legacy Pack ship? If so, you still need to claim the free ship to progress the mission. But no, it's not broken, because I completed it earlier today, without any issues.
IIRC, you also need an empty ship slot to claim the ship. If your ship slots are full, for whatever reason, it can't complete.
STO is about my Liberated Borg Federation Captain with his Breen 1st Officer, Jem'Hadar Tactical Officer, Liberated Borg Engineering Officer, Android Ops Officer, Photonic Science Officer, Gorn Science Officer, and Reman Medical Officer jumping into their Jem'Hadar Carrier and flying off to do missions for the new Romulan Empire. But for some players allowing a T5 Connie to be used breaks the canon in the game.
Someone mentioned this earlier so I took a look at the bug reports from Tribble.
You are not supposed to claim the Dhelan until after you have picked an ally. Doing so can sometimes TRIBBLE up the quest progression. This usually happens when you reach Level 10 before you chose an ally.
Same thing happened to me. I opened my ship from the Legacy box instead of going through the Requisitions contact. I had to go back and accept the one from here and I was able to complete the mission.
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You are not supposed to claim the Dhelan until after you have picked an ally. Doing so can sometimes TRIBBLE up the quest progression. This usually happens when you reach Level 10 before you chose an ally.