TL;DR The English strongly supports the reading that the Word, not the believer, is the subject of "would do what he said". That's because we have parallelism between the verbs in the sentence.
There are two ways we could break down this parallelism: into two or into three parts.
As two parts:
[1 whoever did want] him, [2 who believed] he was who he claimed and would do what he said
As three parts:
[1 whoever did want] him, [2 who believed] he was who he claimed and [3 would do what he said]
The three-part reading does not work. It is defective. This is because proper parallelism must have a consistent structure apart from the element that changes. Concretely, it's the repeated subject pronoun whoever and who that signal the beginning of an item in the list. If we had three items, we would be missing the repeated subject pronoun for the third item.
Since Eugene Peterson (or his editor) is a good stylist, he added the "who" of "who believed" — which could have been omitted — in order to force the two-item parallelism and prevent the three-item parallelism.
Compare this version without "who":
whoever [1 did want] him, [2 believed] he was who he claimed and [3 would do] what he said
If we don't repeat "who", then each parallel item is now only a verb and not subject + verb, which leads to ambiguity of the subject.
Note that a poor stylist, of whom there are many, could easily screw up the parallelism and write either version if they meant a three-item parallelism. Hence, if it were in your average blog post on the Internet or quickly written newspaper article proofread by a proofreading mill overnight, I would not bet on this interpretation. But a world-renowned Bible translation by a gifted writer and seen by many eyes before publication is likely to be more carefully edited and use parallelism intentionally to disambiguate.
By the way, the original Greek is not ambiguous. That's because it reads "as many as did accept him, those who believe," etc. with both the pronouns and the following verbs conjugated for plural, not singular. So Peterson's decision in English mirrors this.