I'm not really sure what Azure has to do with this but assuming you are using SSIS then you are creating a package that ultimately gets deployed (to Azure). To do that development you would be using Visual Studio locally. Assuming that when SQL was installed that the SSIS components were also installed then VS should work just fine. You don't need admin privileges to debug a package. It runs the underlying host process (DtsHost if I recall correctly).
Of course you also need to have access to the SQL server instance you are targeting (for queries) but I assume that isn't an issue here.