3D surface reconstruction can be challenging. I would first recommend taking a look at PCL. The Point Cloud Library has grown into a nice set of tools for 3D point management and interpretation, and its license and API sound compatible with your needs. The surface reconstruction features of the library appear to be most applicable. In fact, RBF reconstruction is supported.
If PCL doesn't work, there are other options:
- MeshLab,
- This SO post provides a nice summary, and
- of course, Wikipedia provides some links
Finally, you might search CiteSeerX, Google Scholar, etc. for papers like this one. As an example, a search for "3D Surface Reconstruction" at CiteSeerX yields many hits. RBF-based reconstruction is just one of many methods: is your application truly limited to radial basis functions? If not, there are many choices, (i.e. Ball Pivoting Algorithm). See this survey paper for some comparisons.