in French, you can use the same word : apparatchik, but in French, this word indicates somebody without a public role. " Un apparatchik de l'UMP " would not be a major political figure. It's even a very negative word within parties : " un apparatchik " has made his career in party's administration/organization and has not been elected at a high level (that's what it says in Russian " apparatus' men ").
Another useful (but old-fashioned) idiom would be " caciques " to name the everlasting leaders of a party.
In French PS (Parti Socialiste = Labour), caciques are named " Eléphants ",
In former RPR (now UMP = Tories), caciques were " Barons ".