Hi Cameron,
I am pretty sure you can handle everything in Asana and enjoy the benefits of an unified view and no trouble and friction while switching between and maintaining multiple tools.
For instance, when people file a request you can use Wufoo to create a task out of this form/request.
I blogged en detail how you can convince your colleagues in your company for Asana here in the Community.
Good luck and welcome to Asana,
Sebastian