Using an ASP.NET application, server-side wise, is pretty much similar to any other .NET application I guess. I have a code sample here running a SSIS package (SQLServer 2008), but I cannot copy'n'paste it here since I'm on a NDA in this project.
But I can tell you, we are using a namespace called Microsoft.SqlServer.Dts.Runtime.Wrapper, interfaces Package and Application and DTSExecResult enum... and it works fine.
And we are importing an Excel sheet to our database as well.
Edit:
Here's the overview of how we do it here:
We have Excel sheets that contains data we want to import into our system. Our system is ASP.NET 3.5/Castle Project/SQLServer 2008 powered.
Once the user uploads a new sheet into your system, we store it in a folder, let's say "c:\UPLOAD". An action is triggered, so we call our SSIS package to import that into SQL Server 2008.
How we implement it (C#):
Package package;
Application app;
DTSExecuteResult packageResult;
String packagePath = ""; // You have to get your physical path to your package.
app = new Application();
app.PackagePassword = "password"; // We have it here..
package = (Package)app.LoadPackage(packagePath, true, null);
packageResult = package.Execute();
All those classes are from the same namespace, Microsoft.SqlServer.Dts.Runtime.Wrapper, as I stated before.
If you are having access problems, I guess you better ask your DBA to raise your permissions or even create a new one just to run SSIS's packages, since it requires additional levels of permissions.
Hope this can help you.