A Pull Script is a script where the data is retrieved/pulled by the vendor. The vendor writes code to the actual AssignmentPro client site. When the client places a service order, the order tells the vendor they may "call" that specific order. The vendor's code then retrieves the agreed data.
A pull requires an enhanced permission set update, as the vendor will require additional access to tables and fields. The benefits of a pull script are the vendor typically does most of the mapping work. The drawback is the vendor will encounter errors if fields are not mapped precisely, and permission sets are more difficult to maintain resulting in monitoring error reports for CRUD errors.