A UK client needs to pay approx 113 EUR for my services. I am registered as a freelancer here in Germany, is it ok to issue an invoice without any sort of tax ID number,apparently they do not even have a self employed ID number.
You should talk to your tax adviser. If you invoice a client from another EU country, you have to submit a "recapitulative statement" (Zusammenfassende Meldung). For this statement you will need the VAT number of your client. More information can be found here: http://ec.europa.eu/taxation_customs/vies/faq.html#item_9
This will probably not be necessary if your client is not a company but a private person. But even in this case, contact your tax adviser! There is also the question of whether you have to add VAT to the invoice total, which depends on the kind of service/product your were selling. More can be found here: http://europa.eu/youreurope/business/vat-customs/cross-border/index_en.htm
If you are just writing an invoice to a client in another EU country without knowing the regulation you are most certainly doing it wrong.
If you have the UK company name and address, you can find their details at https://www.gov.uk/government/organisations/companies-house if they are an incorporated entity (ie, if it's not just a person who is self-employed).
A UK company will not have a 'tax number' as such, but they will have a Company Number if they're incorporated (ie, if their name ends in Limited, Ltd, plc, etc). You can find that at the above link. They may have a VAT number, which you will absolutely need if you are claiming back VAT.
If it's a person who is self-employed, all the information is irrelevant. They may have a VAT number, or may not. Just use their name and address.
Just so I can confirm...
A UK Company owes your German self money?
If true, then correct, they do not need to provide anything. They should invoice you (meaning, they should send you a bill - a simple word document will suffice, with their name, address and bank details on it). Make the payment electronically so there is an audit. Not in cash, the German taxman will think you made a false invoice and pocketed the cash.
Tell your customer that you need it in case you get audited for tax. Be pleasant so you can both avoid misunderstanding. Some Brits will think your request is eurocratic but in reality, you are both right. Your customer does not know German requirements, nor do you know British norms.
If you are paying out regular sum's of money then additional paperwork would prove helpful in case the German tax man audits your books. But 5% or less of your yearly revenue paid into another European bank account should be acceptable.
I understand your concern - remember going forward if you have future foreign relationships that you get a copy of something that says who they are. It can be a photocopy of an ID card or drivers license for example, or their memorandum and articles (every UK company has one). Always pay electronically so there is an audit trail. If you dont get an invoice, create your own so that the tax man knows you are doing your best to ensure an audit trail is kept. That way, if you do do something wrong, they'll just ask you to pay what is owed without a fine.
I hope that helps - best of luck.