Generally yes - the potential charges pertain to attempting, arranging or facilitating the commission of an offence. Such an offence does not necessarily require a 'child' to exist. For England and Wales, this Crown Prosecution Service article seems comprehensive: https://www.cps.gov.uk/legal-guidance/vigilantes-internet-cases-involving-child-sexual-abuse
There are some caveats:
There's no defence of entrapment as such, but in E&W the CPS says some cases have been successfully defended on the grounds that the vigilante actively encouraged the suspect's activity and therefore key evidence was excluded by the court.
The nature of the relationship between the police and vigilante might mean the vigilante has become a Covert Human Intelligence Source within the eyes of the law and therefore should have been regulated by the Regulation of Investigatory Powers Act (RIPA) 2000. In which case the absence of a RIPA authorisation might lead to key evidence being excluded by the court.
The vigilante might have sufficiently misconducted themselves so as to cause an 'abuse of process'.
The vigilante might have involved a child and could himself have committed a criminal offence, or the involvement of the child was such that it would not be in the public interest to prosecute the accused because to do so would encourage other vigilantes to involve children in that way.