In continuing to preach through James, and today discussing speech acts (in debt in particular to J.L. Austin) in the context of James 3, I recollected reading Dale C. Allison, “A liturgical tradition behind the ending of James” JSNT 34 (2011), 3-18.
Allison, with reference to James 5:13-20, suggests that a “very primitive church order” lies behind this part of James. Quite what he envisages a church order to be is less clear; though he does have some reference to prayers in Testamentum Domini, and to Constitutiones apostolorum and to some material the Didascalia, he also cites a number of other early Christian texts, including Polycarp and I Clement, in support of his case.
I think I would say that rather than being influenced by a church order, the epistle and the church orders draw on the same fund of catechetical material.