I doubt whether James is quoting Paul. It is far more likely his words were inspired by the encounter recorded in Luke, when he and his mother and brothers were trying to make their way through the crowds to see Jesus:
19Then came to him his mother and his brethren, and could not come at him for the press. 20And it was told him by certain which said, Thy mother and thy brethren stand without, desiring to see thee. 21And he answered and said unto them, My mother and my brethren are these which hear the word of God, and do it.
-- Luke 8:19-21 (KJV)
It is unlikely Jesus' words didn't make it back to James and his brothers, which would have caused them to seriously reflect on whether they were doers of the word, or hearers only.
The crowd they encountered had just heard Jesus relating the parable of the Sower in which Jesus depicts the word of God as a seed that will produce fruit only if it manages to find its way to the good ground at the core of one's being. "The Law", written on scrolls and parchment, only becomes "the Word" when it speaks to the heart of the one who hears/reads it.
If one were to do a search for "the word of God", it will be shown to be predominantly a Greek expression. The equivalent OT Hebrew expression is "the word of the LORD". Looking through all these search hits, it becomes obvious that "the word of God" or "the word of the LORD" is not something written on a scroll or a parchment, but something that "came unto" the one who received it from God himself, i.e. it was "planted in his heart" just like the seed in Jesus' parable, and it moved him to live it and preach it.
Both James and Paul use the word ποιηταὶ, given in the KJV as "doers", which comes from the Greek ποιητής (Strong's G4163 - poiētēs) and relates to the arts, i.e. to be a performer, one whose actions are governed by the words of the script he has learned.
This sense of the word led to an investigation of the history of acting, which it seems was well developed in Roman culture by the first century AD, having started some centuries earlier with a Greek, Thespis of Icaria:
Before Thespis, the chorus narrated (for example, "Dionysus did this, Dionysus said that"). When Thespis stepped out from the chorus, he spoke as if he was the character (for example, "I am Dionysus. I did this").
-- The History of Acting (Wikipedia)
So, James and Paul would have their listeners be Thespians, i.e. those who "want to" step out of the chorus and "become" what is written.
I get this from the Greek of James 1:21-22,
21 Therefore, being divested of all depravity and profusion of malice, with meekness receive the infused word (that which is able to save your souls) 22and become doers of the word and not self-deluded hearers only.
It should be noted that the KJV translators have not given the word James uses (γίνεσθε) any differently than they would, say, είμαι ("to be"). But, James is using strong language here: having put off the vestments of depravity and malice and with meekness taken hold of the infused word of God, BECOME DOERS of the word (something that previously YOU WERE NOT), and not just SELF-DELUDED1 hearers.
"The word of God" is that which is spoken directly to the heart of a person and it comes by means of the words of God read or heard. So, a "doer of the word" is one who acts out the script as he has received it from God. A "doer of the Law" is no different, and a "doer of the word" will become a "doer of the Law" since the Spirit of God will move him to want to learn from all the words God spoke to the fathers and to Moses and to the Prophets, to Jesus and the Apostles.
Jesus says this:
He that is not with me is against me; and he that gathereth not with me scattereth abroad.
-- Matthew 12:30 (KJV)
Everyone takes a role in the drama. James and Paul would have their listeners learn their script from Jesus. Jesus is saying the part one is to play is that of a gatherer, just like him. All other parts are variations on the role of scatterer.
Paul's words in Romans 2:13 suggest that he is taking a James-like look at the word, rather than the other way around, i.e. James taking a Gospel-of-Grace-like look. Perhaps one might find such a thing in some other part of James' letter, but it doesn't appear so here.
- ἑαυτούς is person-less, i.e. it refers to "selves", be it "yourselves", "ourselves", "themselves". So, παραλογιζόμενοι ἑαυτούς stands here as an additional adjective ("self-deluded") modifying the noun ἀκροαταὶ ("hearers").