am I required to complete the ESG questionnaire from my customer

No legal obligation, but a contractual one

No, there is no law that says you must fill in a customer's ESG questionnaire. The CSRD (Directive 2022/2464) requires certain large and listed companies to report on sustainability — your customer may fall under this, but you as a supplier usually do not. What does happen in practice: that customer translates his CSRD obligation to his entire supply chain, and asks you for the data he needs himself. This happens via the agreement between you, not via the law. If you refuse, you are not breaking any rule — but the customer can weigh it into the relationship, just like any other contractual request.

Many SMEs therefore fill in the list via EFRAG's VSME baseline module, the voluntary standard that is tailored precisely to non-listed companies. That standard is not mandatory, but it is useful: it covers the questions that large customers usually ask, in a format that already exists. This way you don't have to think about how to the number of employees of the undertaking or the countries where the company has establishments formulate each time — those answers are already fixed and come back in virtually every ESG list.

Where that difference comes from

The CSRD (Directive 2022/2464) places the reporting obligation on the company that falls under it, not on its suppliers — the supply chain information a customer needs, he obtains via the agreement with you, not via a direct legal obligation for you. EFRAG's VSME baseline module is explicitly voluntary and designed to standardize that supply chain information, so that an SME supplier does not have to come up with a different format for each customer.