I've had the theory for years now, but I've started leaning more on the "No" side.
Unfortunately, Lee Unkrich denied it with a simple "No" on his (now-deactivated) Formspring account. But while we can't rely on what's no longer there, let's examine.
In matters like this, I tend to go with Pixar's word on things. So if Pixar says his (absent) father passed down Woody to Andy, then that's what I believe happened. I'm sure John Lasseter has mentioned this in passing a few times.
The hair is a matter of might be-might not be, of course. Her hair in Toy Story 2 and 3 is more akin to what Emily's hair looks like, but the one thing that makes me question it is her hair in our (brief) view of her in Toy Story 1.
And there's everyone's argument of the hat being the same color, but Andy's hat and Jessie's hat are completely two different shapes, whether or not Andy's hat has a faded band. Andy's hat, while far from color to it, is shaped more like Woody's hat; round triangular shape. Emily's is basically a replica of Jessie's; circular.
But despite all this, this isn't to say she isn't Emily. Of course, hair dye is a thing. Maybe she loved that Andy's father owned Woody (I'm sticking to this belief until Jossed by Pixar or Lasseter). And maybe she bought Andy a red hat because it reminded her of her own. So for now, I'm sticking on the more "Unlikely, but possible" side of things.