Readers’ forum: Should UK-based computer hardware seller charge VAT?

My client (a UK-based and VAT-registered sole trader) has bought some computer hardware (a new keyboard) online. I have suggested that the client obtains a VAT invoice. The supplier has responded with an invoice that says that the supply is outside the scope of VAT and that therefore no VAT has been charged on the supply. The supplier is UK-based and has a UK VAT registration number. My client does not remember stating that he was purchasing the keyboard for business purposes although that was indeed his intention.

I have three questions. 1: Is the supplier correct not to charge VAT?  2: Should the reverse charge mechanism apply? 3. Assuming VAT was payable on the transaction HMRC could presumably seek input tax from the supplier (one-sixth of the purchase price). On that basis can my client claim one sixth of the purchase price as input tax (using the non-VAT invoice)...