After speaking to Commander Kelly, he tells you to go get a phaser rifle from one of the nearby crates.

I have explored the entire nearby area, and none of the crates are interactive objects - there's just nowhere to pick up the rifle from!

If it helps, I did experience a game crash just after beaming down to Vega Colony. After re-joining the game and picking up, I can't find the crate.

actually the reason you can't interact with any of the crates is because so many people are trying to activate these crates at the same time. if you hang about eventually you get the option to activate and if you're quick you can nab it.
however, inspite of managing to activate, it won't let you 'take' or 'take all' so you still can't retrieve the rifle. i hung about for quite some time and tried several times and same story.
and the situation was the same for pretty much everyone as far as i could tell (apart from the poor buggers who complained in chat of appearing as their ship!)

ditto

spent a good 30mins cruising that area, I opened loot window a couple times.
neither times did i get item, it just hangs.

Same issue - bascically in the last 30 minutes before the server was shutting down for the 7 pm PST patch, you could not finish the tutorial because you could not aquired a phaser rifle. It might evenutally spawn, but it was impossible to pick up.

I had the same issue, I changed instances and was able to retrieve a rifle.

Well i have an issue with the vega colony, I selected my away team and they never beamed down with me

I think I was able to get one, but I must have dropped it by mistake because it isn't letting me pick up any more from the crates, the NPC is unavailable to talk to, and I can't progress in the quest because those things have to happen.

What should I do here....start over completely with a new character? Surely not...

Any ideas; bump for great justice?