Since $\mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $\mathbb{S}^3$.

If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $\mathbb{S}^3$ is a composition of inversions ([Liouville's theorem)][1].

So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.

The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths. 
Since $M$ is simply connected it gives a well defined conformal immersion $M\hookrightarrow \mathbb{S}^3$.


  [1]: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)