In the UK, domestic work is VAT-exempt if you don't meet the income threshold for VAT (currently £82,000).
If they ask for your VAT number for tax purposes, simply state that you are not registered for VAT and do not charge VAT. If they complain, they have a mediocre accounting department as this is the very first section on VAT in almost all documents describing it:
You can only charge VAT if your business is registered for VAT.
So no, you must not add VAT on your domestic UK invoices to businesses unless you have a VAT number. On the invoice, state "VAT Not Applicable" or something to that effect.
There is also no need to register with an umbrella company (agency) for this purpose, either, unless you would like to use their insurance, company name, etc.