I see that EBay has an affiliate API for sending customers to EBay: http://developer.ebay.com/devzone/guides/ebayfeatures/Basics/eBay-AffiliateTrackingConcepts.html
However, is it possible to track conversions and see if a specific customer purchased a specific product? I can't find any data on getting affiliate reports or what the reports contain.
If you go to the eBay Partner Network and click on Reports > Download Reports you can generate reasonably detailed traffic and revenue reports, down to the ItemID.
With regards to determining which specific eBay username has lead to the conversion, I really don't think this is something that eBay would make publicly available to you.
As much as we would all like to know who it was it was that entered into a frantic bidding war at 23:06:11 to win that almost new Jenna Jameson Moulded Pussy and Anus, in many circles this is considered personal information.
The above notwithstanding, however, you could apply your own tracking methods through the use of advertising campaigns, which are trackable within the revenue reports. Batch campaigns can be made by clicking Campaigns > Create Campaigns in Bulk. From there, it would be a matter of disseminating links to each campaign and logging to whom each was advertised.
More information can be got from the User Manual