Got a query. For a company that deliver material to customers, when the vehicle is in the other Companies Yard, and their employee gets on the back of the Vehicle to off load some material and falls off. Is the company that was delivering the material at fault for allowing the person on the vehicle

Could be either or both. In this situation, joined up planning is needed. Very limited information so a definitive answer cannot be established!

What lead to the employee falling? Was this task planned for and risk assessed? were these controls followed? Was it due to a defect in the trailer? or defect in the load / off loading task? etc. etc.

Are you asking about fault due to an insurance claim? For accident reporting reasons it is clear that the employer reports (if this is needed under RIDDOR). For insurance purposes, again the claimant will go after the employer in the majority of instances (as this is generally easer for them).

Lots of questions, not many answers :-)