You can start selling your service(s) without a VAT number, but it's obligatory to register your company as a VAT payer if:
Your turnover reaches €40 000, from the beginning of a calendar year (an exception applies when selling EXCLUSIVELY outside the EU) AND/OR
You start providing 'digital' services (software, digital content etc) to individuals or companies in the European Union without a valid VAT number based on VIES database (meaning that your company has to be registered for the MOSS Scheme)
If neither of the conditions mentioned above are met, applying for VAT-payer status is voluntary. There are some instances in which applying for a VAT-payer status may not be a good idea.
For example, you might not want to register as a VAT-payer if your customers don't have VAT numbers. This will be the case if you plan to sell your services to:
only individuals in any of the EU member states OR
only companies and individuals outside the EU
to a company that is registered in one of the EU member states, but does not have a valid VAT number (you can check the validity of their VAT numbers from the VIES database)
When deciding whether to register your company as a VAT payer or not, you should also consider your expenses, as the location of your potential customers and suppliers impacts the result from a VAT perspective.