This plugin works with Blender 2.79 only.

Modified obj exporter that adds vertex group-weight mappings
to vertices. This extension makes it possible to use the obj
format for models that use armature deformation (skinning). 

See "export_obj.py" for a more detailed explanation.

A custom obj importer is required to read the vertex groups
and weights. However, they will be recognized as comments 
(and thus ignored) by regular obj importers, so the files 
should still be compatible with them.

To install, open Blender user preferences and select install
from file, then select this zip file.

Alternatively, put the io_scene_oddwarg_obj folder into:
"Blender\<version>\scripts\addons"

Importer which preserves weighted vertex groups was provided
by ntfwc, with the following note:

When enabled, this adds a new import option "Wavefront, with oddwarg extensions". It should work a lot like the standard import option. Note that "Keep Vert Order" is default. Using "Split" will scramble the vertex groups, and should be avoided, at least for now.