Hi,
I think you can do your scenario in two parts:
1. Http (or Rest) - File. With SOAP Axis adapter (Michal's PI tips: Exchange Rates from an XML file on a web page - REST, AXIS) or Rest adapter (PI REST Adapter - Polling a REST API) you can do a poll to search in the website for file at every X minutes and to deposit the file.
2. File to ECC Proxy. You do a poll to send the file to the ECC proxy. At mapping level, when you know securely the file has being processed you can do a call to delete the file on the website (With SOAP call or a call similar than this Yes Rest LookUp is possible in PI).
Regards.