Is there a single word for a member of a group who completes a job?
Solution 1:
All of the -er words are fine examples, but I can't think of any use of them that won't seem awkward.
Why not use "done by", "completed by", "finished by" or a similar construct?
Solution 2:
Completor may not be a word, but completer is. Here's the Wiktionary entry for it; it can indeed be a noun meaning 'one who completes'.
Solution 3:
You might try finisher, which suggests "[bringing] a task or activity to an end" [Noad].
Solution 4:
You could go with achiever (would be my personal choice), finalizer, or fulfiller.