Vayne solo'd baron while taking a teemo shroom without full build or any defensive items

That isn't possible.

Doesn't seem likely considering even champs like Udyr have trouble soloing Baron

I call bs. She'd need at least 5 items and an Elixir of Wrath.

What you're really saying is that Vayne didn't yet have her 6th, situational/defensive, item yet and she got hit by your, probably, on-hit AS-AP Teemo shroom that does jack shit past 30 minutes.

Pics or didn't happen.

She would need sated and full lifesteal to even be close

Show me.

What was Vayne's build? I mean, I could see it if she had something along the lines of: (possibly too)

A lot of Baron's damage is AOE and avoidable, so she could have just dodged a lot of it by walking around.

Nevermind, found the game. The Vayne had 31 kills and 7 deaths by the end with the most CS in the game. She was just fed out of her mind and everyone on your team was behind in gold by a massive amount.