Generally access to the Programm folder is limited at best. There used to be a time when we even would write data like Configuration files there. That was Windows 95/98. And it was frowned upon back then.
With the introduction of NTFS with Windows NT (and later XP in the Consumer area), enforceable rules were put in place. Rights tend to be highly limited. This is even more true of Webservers, as they tend to be prime targets for hacking. The only rights you can expected a Webserver to have is read rights to it's programm and the content directory. Maybe if the admin is really nice and it really helps with performance, it might have write rights on a Temporary subfolder of Contents.
Reading out your version from a file is propably possible with only read rights, but not the right way. Usually you just let the compiler hardcode (insert as constant) whatever Version is given in the Project file or by the Version Tracking Software.
So the first thing is that you need some approach of Version Control. And then your answer depends "however this Version control mechanism does tracking of the Compiled version". There is some limited version tracking support in Visual Studio itself. But there is also full, 3rd party Version tracking Software.