You could make use of either
blob storage or
file service for that purpose.
Since you're migrating an existing application, I would recommend going
with azure file service. Basically what you will do is create a share and
map that share as a network drive on both web role and worker role when the
role starts. Web role will then write a file to this share using standard
file API. To notify the worker role that a file is there to be processed,
web role could write a message to a queue which is being polled by worker
role. Upon receipt of the message, worker role will read the file and
process that file.
You may find this blog post useful regarding azure file service: http://blogs.msdn.com/b/windowsazurestorage/archive/2014/05/12/introducing-microsoft-azure-file-service.aspx.