I believe the answer is (in general) <b>No</b> for the following reason.  A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic.  This was asked by Ringle in 1956, and answered by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$.
I find this explicit claim in a 1988 paper by Suvorov ([Springer link][1]):
<br />&nbsp;&nbsp;&nbsp;![Suvorov][2]<br />
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).


  [1]: http://www.springerlink.com/content/q024m7649g513x04/
  [2]: http://cs.smith.edu/~orourke/MathOverflow/Suvorov13.jpg