Should be possible. I've beaten the Vanilla Destiny Story up to the Black Garden with only Starter Gear, although it became pretty hard at the end.
However, I've done this in October 2014. Since then the game has changed a lot. It may be much more difficult now. Actually worth a try...
[spoiler]Though I am certain you cannot beat The Taken King with only Starter Gear. I've played with the Khvostov a few times on Patrol The Dreadnaught, and all enemies take a ridiculously long time to kill.[/spoiler]
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I'm impressed you understood the question
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This is actually, the best answer yet. You get a gold metal, heck, you get the All-medal.