Microsoft Visio supports many vector formats for export purposes, including the SVG format (XML based graphics). You can access this function via the default export function. Since shape files from Visio (.vss) are rarely supported outside the program, this is probably one of the few (maybe the only) viable choices for this conversion.
Convert Microsoft Visio smartshapes data to XML based scalable vector graphics format.
We have found one software suitable in our database for this conversion.
Microsoft Windows software - convert vss to svg on Windows
A vector based diagramming software from Microsoft
Convert from vss file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.