How to create a renewal request from the subscription page?
If I understand your question correctly, you are using the Entity reference module on you subscription content type to point to a (new) node of the type renewal_request.

The Entity connect module extends the Entity reference field by providing an add and edit button:


  The "add a new" button will enable you to add / create a new entity. The first  form will be cached then the new form will be generated. Once the new form is  completed and saved the user will return to the original form.
  
  The "edit" button will take the user to the edit form of the referenced entity. Once the user is done editing, he is sent back to the previous form.


You could also use the Inline Entity Form module:


  Provides a widget for inline management (creation, modification, removal) of referenced entities.