"From 1 January 2015, telecommunications, broadcasting and electronic services will always be taxed in the country where the customer belongs* – regardless of whether the customer is a business or consumer – regardless of whether the supplier based in the EU or outside."
Ref: EUROPA: Taxation and Customs Union - Telecommunications, broadcasting & electronic services
If you register on their website as a business customer then VAT is payable in the EU country where the customer belongs (UK in this case), if you register as an individual customer then VAT is payable in the EU country where the supplier belongs (Germany in this case).
Having tried that site just now, the EU VAT registration number field appears to be optional though and not mandatory.
I have noticed that some websites for a while now have used the VAT registration number to verify if a customer really is a business customer since they were only required to collect VAT for B2C sales only and required B2B customers to self report under the reverse-charge scheme for EU VAT. My understanding (please someone correct me if I'm wrong) is that EU VAT applies to all international sales within the EU, both B2B and B2C now.
Added on 01-Feb-2016:
The guidance that would apply to this supplier is as follows:
"If you supply digital services and your customer doesn’t provide you with a VAT Registration Number (VRN) then you should treat it as a business to consumer supply and charge the VAT due in the customer’s member state. If a customer is unable to supply a VRN but claims they are ‘in business’ but not VAT registered because, for example, they are below their member state’s VAT registration threshold, you can accept other evidence of your customer’s business status eg, a link to the customer’s business website or other commercial documents.
It’s your decision whether to accept alternative evidence that the customer is ‘in business’ and your customer can’t require you to treat a supply as business to business if they haven’t provided a valid VRN.
If you accept that your customer is in business, the supply doesn’t 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 member state. You must complete and submit a quarterly European Community Sales List declaration to HM Revenue and Customs (HMRC). Other EU member state tax authorities are then able to request details of the database where these declarations are securely stored for taxpayer compliance and audit purposes."
Ref: GOV.UK: VAT - businesses supplying digital services to consumers
Essentially this means they shouldn't be requiring you to provide a VAT ID but that if you don't specify one, they can either choose (if they wish) to accept proof of your low turnover for VAT registration exemption and deal with you as a business keeping this evidence to satisfy their tax obligations and accountants, or alternatively deal with you as a consumer and charge UK VAT on any orders. There is no legal requirement I can find for you to supply a VAT ID in order to become a customer of theirs.
I would have originally thought this would have probably been an error in the webdesign of the registration form treating it as a required field when it shouldn't be. Also, they may have misunderstood the VAT rules? A conservation by phone or email between them and yourself may be required to proceed further here, you could reference the links I've added to this answer.