How do I take over an inactive fanclub?

You can't "take over" a club. Clubs aren't owned by the person who started them - they're owned by every member. However, you can still get the icon and banner changed. The method is different depending on whether the club has 10 or more members.

The first thing you'd need to do is start a forum thread in the club asking people to suggest new icons and banners. Leave that running for about a month or two. Then,if the club has less than 10 members, you make 2 polls asking members to vote on their favourite icons and banners, then message one of the F4 asking them to change the club's appearance to the winners. If the club has more members, you need to use the buttons on the top right of the club's main page which say "suggest a new banner/icon". These make polls which cost members 5 golds to vote (to discourage uninterested people from spam-voting) and the icon and banner get automatically changed to whichever win these polls.