How do I obtain the actual work records showing that a properly trained and certified company performed the required work?

This may be more difficult outside the context of litigation. You should request such records in writing from the landlord. Request that they maintain such records of purported compliance with the turnover provisions of Local Law #1 of 2004 for the complete duration of your tenancy.

Beyond that, it is prudent to consult with a lawyer who will give you guidance on any right which you may have in obtaining such lead abatement work records. Obviously, such records may be evidence that the required work was actually performed by appropriate contractors and completed correctly. As a tenant, you are not in a position to be aware of the landlord’s compliance with the law absent access to such records.