If you are certified to ANSI/ESD S20.20, you can claim and you must in your documentation that you can handle devices that have a withstand to 100 v Human Body Model (HBM), 200 v Charge Device Model (CDM). A process classification does not exist to say your process is a Class 0 process.
Some questions back to your supplier. What is the withstand voltage of the devices you wish for us to handle? What do they mean by Class 0? In the device world, Class 0 applies to HBM only and goes down to 0 v. In fact, the new joint standard for HBM has 0Z, less than 50 v, 0A 50 to 125 v and 0B 125 to 250 v so even asking for Class 0 has no meaning.
For CDM the classifications are C0a less than 125 v and C0b from 125 to 250.
With the certification to ANSI/ESD S20.20 your process in in the 0A for HBM range and in the C0b range for CDM.