no general contractor

Alexander - I've never had a situation like you describe with no general contractor. It sounds somewhat problematic to track your overall waste and have confidence it is accurate, which would make credit achievement questionable. When in doubt always come back to how you are meeting the credit intent, which in this case is "To divert construction and demolition debris from disposal in landfills and incineration facilities. Redirect recyclable recovered resources back to the manufacturing process and reusable materials to appropriate sites."

Here are a couple of things to consider:

1. Someone will need to take responsibility for tracking all the waste generated on the job as well as what materials are recycled (and where) vs. landfilled.
2. Your tradesman can measure their waste by volume instead of weighing it to get an accurate accounting (see LEED Reference Guide or https://leeduser.buildinggreen.com/credit/NC-v4/MRc5 for a table of conversion factors). You might also review the FAQs for this credit - https://leeduser.buildinggreen.com/credit/NC-2009/MRc2#faq. 
3. To get an accurate amount for the LEED project, they cannot mix the waste from the LEED project with the waste from another project before they measure its volume.
4. You'd need to get the tradesman to account for whether the measured volume of material is diverted (and where) or landfilled.
5. What are you doing about large amounts of waste on the job (what tradesman aren't taking back to their yards)?