Well, it sounds like you're experiencing the "double-hop" issue. If you do not have Kerberos authentication enabled between these servers on your network (I do not know much about this) you cannot pass the credentials of the web client user to the web server (first hop) and then to the SQL Server (second hop).

One solution would be to create a standard SQL login with the appropriate permissions to execute sp_start_job. Use this login in your asp.net connection string to call the stored procedure that executes the SSIS package.

It is not ideal, but it works. However, the job is always called by the sql login that was created, so you do lose the audit trail back to the actual user.