No, the ability to bring a private prosecution is not restricted to specific entities - it is a right everyone has (though some offenses can only be prosecuted with the consent of the DPP, and the DPP can choose to take over a private prosecution) [1]. For example, a woman tried to crowdfund a private prosecution for rape [2].
From [1]:
> There are a number of organisations that regularly prosecute cases before the courts of England and Wales but they do so as private individuals, using the right of any individual to bring a private prosecution. One example is the RSPCA.
[1]: https://www.cps.gov.uk/legal-guidance/private-prosecutions