Some domain names are 'restricted', whereas others aren't.
For example only recognised schools, colleges, universities and relevant research bodies are allowed to have domain names ending in '.ac.uk'. Similarly only govermnent bodies and local councils can have domain names ending in '.gov.uk'.
However the majority of domain name extensions are available to anyone without any restrictions. So, as long as it hasn't already been registered by someone else, you're free to register 'rosie29.org' as your domain name and then to do what you like with it. (You could use it to run a charitable organisation but, equally, you could run a business with it or host a porn site using it).
The rules about setting up and registering charities can be found here:
https://www.gov.uk/setting-up-charity
However a 'good cause' isn't the same as a charity. Your local primary school's Parent Teacher Association might raise funds for the school as a 'good cause' and it's free to do so without ever formally becoming (or registering as) a 'charity, irrespective of its level of income.
Many independent schools though
are registered as charities, even though some might argue about whether they constitute 'good causes'. (e.g. Eton College has investments worth more than £100m, and charges the parents of its pupils mega-bucks, but it's still a registered charity).