Does this mean I have to register for VAT MOSS, and VAT?
Vat Moss only applies to sales to people who are not in business "I.e "individual consumers" - (B2C sale)
As your customer is a business this is a B2B sale and Vatmoss won't apply and there won't be any adverse vat implications unless your work falls under one of the quirky rules where you would need to register for vat in the customer country - this won't be the case for you (Hmmmmm a bit presumptuous there perhaps)
Note you will need reasonable evidence your customer is in business if they don't supply a vat number that passes verification tests - details in that regard via link provided
How to determine whether the customer is in business (a taxable person) or is a private consumer
If you supply digital services and your customer does not give you a VAT registration number (VRN), you should:
treat it as a business-to-consumer supply
charge the VAT due in the customer’s EU member state
If a customer cannot supply a VRN but claims they’re in business but not VAT-registered because, for example, they’re below their EU member state’s VAT registration threshold, you can accept other evidence of your customer’s business status. For example, a link to the customer’s business website or other commercial documents.
It is your decision whether to accept alternative evidence that the customer is in business and your customer cannot ask you to treat a supply as business-to-business if they have not given a valid VRN.
If you accept that your customer is in business, the supply does not come within the scope of these business-to-consumer arrangements. With a cross-border business-to-business supply the customer will be responsible for accounting for any VAT due to the tax authorities in their EU member state