I'm not a new player, however I've been focusing on playing the original class stories and only reached max level for the first time recently. Since then, when I've done planetary heroics, I get a "locked supply crate" as a reward, which apparently is supposed to be turned in to an npc in my alliance. The problem is that I didn't play through any of the expansions so I don't have an alliance. Is there any other way to open them, or will they just remain cluttering up my inventory until I get around to playing whichever expansion gives me the alliance?

That is their only function. They only have value after chapter 9 of KotFE, when you can give them to the specialists.

 

(I suspect the answer is no, but check if they can be sold on the GTN. I think they're bound to legacy though.)

You can't do anything but store the crates them until you complete chapter 9 of KotFE.

 

Auto-complete of chapter 9 counts though. If you have a level 65 character, and don't mind having storyline auto-completed you can jump to KotFE chapter 1, exit after starting it, and then jump straight to KotET chapter 1. That will give you access to the Odessen base and you'll be able to turn in the supply crates for goodies.

 

You can also take a level 70 character and jump straight to the Ossus storyline from your ship's mission terminal. That will also auto-complete the storyline and give you access to the Odessen base.