This is a great question. The chief point to keep in mind is that the indefinite relative pronoun is identical in usage to the relative pronoun. In other words, the case of indefinite relative pronoun is determined by its function in the relative clause and not the sentence.
In this case, it is clear that the subject of the relative clause is the word, "they." In other words, "they" are the one who are "feeling." In light of this, the relative pronoun must be in the accusative case, or the case of the direct object. Hence, the sentence should read as follows:
"The judges will consider whomever they feel is qualified."
The main point is to always look at the relative clause and work out what is the subject and object. If you do this, you will be infallible with relative pronouns. Here is one final example:
"The judges, who are impartial, will consider whomever they feel is qualified." The "who" is the subject this time.
"The judges will consider (whoever,whomever) they feel is qualified."
Which one is it whoever or whomever?
To determine how to use whoever and whomever, you just follow some simple steps.
1. First determine what clause it is part of. In this sentence, it is part of
(whoever,whomever) they feel is qualified. (You can ignore the rest of the sentence now!)
2. Next, Turn the clause into regular order.
they feel (whoever,whomever) is qualified
3. Now that the sentence is in regular order, it is easy to see that They is the subject and (whoever, whomever) is the direct object.
4. Recall that who and whoever are nominative (subjects and predicate nominatives) and whom and whomever are objective (all types of objects).
5. The sentence should, therefore be
The judges will consider whomever they feel is qualified.
This will work for you every time you have to decide between who and whom or whoever and whomever.