Is anyone among you sick? Then he must call for the elders of the church and they are to pray over him, anointing him with oil in the name of the Lord; and the prayer offered in faith will restore the one who is sick, and the Lord will raise him up, and if he has committed sins, they will be forgiven him. (James 5:14–15)
It seems to me that the expression, “the elders of the church” is quite precise and technical, referring to the elders as the shepherds of the flock. We see the same expression/phrase in Acts 20:17, with the same connotation. The term can refer to older men (e.g. Acts 2:17; 1 Timothy 5:1), but most often it is referring to a leader (Jewish or church).
Could this instruction be carried out by someone other than “the elders”? Surely there is benefit for others praying for someone who is sick, but the primary focus of the elders should be “prayer and the ministry of the Word” (Acts 6:4). In James 5 the elders have a kind of representative role, it seems, and I’m not sure someone else can fulfill that. Also there is the matter of sin and confession of sin that sounds “elderly / elder like.”
So while it is good to ask godly people to pray for the sick, James 5:14 primarily invites the sick to call on the recognized elders of the church.
As a side note, according to the passage, it is the sick person who takes the initiative to call on the elders, rather than the elders taking the initiative to call on the sick person. By this we mean that it is the faith of the sick person that is important here, not just that of the elders.