Can there be a charge configuration in space such that at any instant of time I can change the electric field at one and only one point?
The electric field obeys Maxwell's equations and in particula the Gauss's law. This means that $\rm div E \sim \rho$. If you change the field at only one point, this will introduce an infinite divergence and consequently infinite density. In other words, you'd have to introduce a point charge. But then the field $E$ itself will diverge at that point. So the answer is no, you can't do this.
In general, the allowed configurations are continuous except at boundaries of objects (where $E$ has to jump to account for charges in the material) and at point charges (where the field diverges).