Essentially you'll need two data structures (logical, intrusive, or real, depending on the rest of your code). The first will track the chains of objects, and the other the path.
Chain Simply you need to know which objects are following other objects. In the simplest case this will simply be A follows B, but could include more followers. There is a designated leader in the chain.
Path For each chain you'll need a path. Depending on how your game works will determine how this is structured. In most cases it will be some kind of linked list. This will track the positions that everybody in the chain needs to follow.
Now, the leader in the chain will be adding items to the path. Each time it moves it will add something to the head of the list. Each object in the chain remembers where on the list it is. When it comes to moving it simply moves to the next item in the list (interpolated appropriately if necessary). As the last item in the chain moves past an item in the list, that item can be dropped (it will be at the tail).
Metaphorically the leader leaves a breadcrumb trail for its followers. The last follower in the list consumes the breadcrumb.
Whether your list contains individual points, or just the vertices of a path, or something else, is determined entirely by your game engine. But in any case I don't see that you'll be able to avoid the list itself.