Would another free app or program be better and easier to do this than mapbox.com ?
Excel can hold the attribute values, but doesn't manage the boundaries (the shapes of the census districts).
I haven't used MapBox before, but a quick look suggests it's only applicable for points (rather than polygon boundaries).
An alternative to using a spatial server is to store the census boundaries in GeoJSON. There's a nice example on the Leaflet site here.
There's a good introduction to some popular free GIS programs available on the OpenGeo site.