It is very possible to convert a vector image into a raster image, through a process known as ``rasterizing.'' This is used to draw vector images in the computer screen, and to print them. Many vector graphics applications will have a ``rasterize'' or ``export'' function, where you can turn the vector graphic into your favorite format for publication on the web. Just remember that this new file will not have the same editability that the vector file did.
Conversion from raster to vector is much more difficult. There exist some programs to asist, but in general this is a tedious ``by hand'' process. It is often easier to recreat the image from scratch.