outside of Adobe and Corel Draw is there an online solution for taking a PNG, JPG or SVG file and breaking it apart to change the colors or styles?
Firstly, I should note that PNG and JPG are Raster, not Vector. See "What are the differences between vector graphics and raster graphics?"
As for a vector editing program:
Inkscape is a free before editing software. Available for Windows, Mac and Linux.
There are also many online solutions, which a simple search for "Online vector editor" will bring up lots of results.