There is no legal requirement for them (or any other employer) to advertise any vacancy either internally or externally, and there is no Charity Commission requirement linked to their charity status either. It's best to do so as like any employer charities are required not to discriminate and the easiest way to demonstrate they haven't discriminated (and also to ensure they get the best person for the job) is by having a proper recruitment procedure. But no specific legal requirement.
However many charities may be required to do this because of restrictions on their funding. Often charities have very robust and open recruitment policies anyway, in order to be able to demonstrate to trustees and donors that money is being spent responsibly and in the public interest. Additionally, if a charity is in receipt of funding from another body, particularly a government body, then it is very common for conditions to be attached to that funding including external advertising of all vacancies.
So the short answer is legally no, but in terms of the charities own internal policies and funding restrictions, quite possibly yes.