I'm not VAT registered in the UK as I don't need to be as my turnover is not high enough. Company registration numbers are only issued to businesses that are registered at Companies House, such as limited companies and Limited Liability Partnerships (LLPs). I am a sole trader and sole traders do not have company registration numbers because they are registered with HMRC, not Companies House.
I have an EORI number, this is a number required by a business to move goods between Great Britain and other countries. It appears on all customs paperwork and is a way to identify the registered name and address of the business. You can use this number to verify my business, surely?
If you only accept VAT registration numbers and CRNs then you'll be putting another massive hurdle in the way of micro-businesses who always seem to be overlooked.