Yes, there is such a model structure, at least if we are willing to pick a category of simply connected spaces that has limits and colimits.
The modern proof simply applies the existence theorem for left Bousfield localizations of combinatorial (like sSet) or cellular (like Top) model categories at a set of morphisms.
Simply connected spaces cannot be encoded by a left Bousfield localization of spaces, since the property of being simply connected is not stable under homotopy limits.
However, if we assume simple connectivity from the start, then it is easy to construct such a model structure.
For example, start with the category of simplicial sets that have a single 0-simplex and single 1-simplex. This category has a model structure with the same weak equivalences as simplicial sets, and encodes pointed simply connected spaces.
The left Bousfield localization of this model structure at the set of morphisms given by rationalizations of (pointed) spheres exists and encodes pointed simply connected rational spaces.
In this model structure, weak equivalences are precisely rational weak homotopy equivalences, fibrant objects are precisely rational Kan complexes, and cofibrant objects are precisely all objects (every simplicial set can be thought of as a CW-complex).
If one really wants topological spaces as the underlying object, we can do the following trick: encode simple connectivity as data instead of property.
This amounts to specifying for every pair of points a specific choice of a path connecting them, and for every closed loop a specific choice of a disk filling out the loop. I haven't checked all the details, but I think the resulting category is complete and cocomplete and admits a model structure transferred from topological spaces along the forgetful functor, which can then be left Bousfield localized at rationalizations of spheres, obtaining a model category of pointed simply connected rational topological spaces.
