I don't think file write access is the problem, since in the medium trust level, file access is permitted within the application's virtual directory hierarchy.
Here is a list of the capabilities and restrictions of the medium trust level, taken from How To: Use Code Access Security in ASP.NET 2.0:
Permissions are limited to what the application can access within the directory structure of the application.
No file access is permitted outside of the application's virtual directory hierarchy.
Can access SQL Server.
Can send e-mail by using SMTP servers.
Limited rights to certain common environment variables.
No reflection permissions whatsoever.
No sockets permission.
To access Web resources, you must explicitly add endpoint URLs—either in the originUrl attribute of the <trust> element or inside the policy file.
Godaddy.com's shared hosting servers use this level with the addition of OleDbPermission, OdbcPermission, and a less-restrictive WebPermission, and they explicity state that "file system access is limited to the application's virtual directory hierarchy": What Is Medium trust level and how does it affect my hosting account?
On my side, I have tried adding <trust level="Medium" originUrl=""/> to my application's web.config file however I get an error stating that this cannot be overridden. I do not have access to change anything outside of my application. If you can suggest anything else that I can do on my end then please do so.
I think it's the components themselves that need to be changed. I have done some research and found that there are various modifications that can be made if they have not already been done.
For example, you may need to mark your assemblies with the AllowPartiallyTrustedCallersAttribute, which should be declared at the assembly level (in Assemblyinfo.cs or Assemblyinfo.vb).
e.g. in C#: [assembly:AllowPartiallyTrustedCallers]
or in Visual Basic: <assembly:AllowPartiallyTrustedCallers>
Your assemblies should also be strong-named.
I have read about other third-party components that have had this error
and with some slight modification and recompile, the error was gone.
Here are some pages that look useful:
Allowing Partially Trusted Callers
Flickr.Net API and Medium Trust
By not fixing this error you are restricting your market. Not everyone runs their web sites on dedicated servers. I hope this can be fixed soon.