If you can securely save in your program username/password/domain of the user in witch context you want to launch the process you can use the Process.Start variant that accept, other than the filename to launch, the login data of the user in witch context to launch the process:
Public Shared Function Start (fileName As String, arguments As String, userName As String, password As SecureString, domain As String) As Process
In your code it will be something like:
Dim proc As New System.Diagnostics.Process()
Dim SS As New Security.SecureString
For Each C As Char In "MyPassword"
SS.AppendChar(C)
Next
proc = Process.Start("Avaya 7.0.exe", "", "NonAdminUsername", SS, "MyDomain")
Check Process.Start documentation for more info: