Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in such a way that order type is preserved (equivalently, no three points become collinear) throughout the operation?

I believe the answer is (in general) No 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 (the isotopy conjecture), and answered negatively 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):
Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) RichterGebert found another
14line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):


