So is this something that we ought to do when a person is sick?
I think it would be helpful to look at the immediate context of the passage you have been asking about:
James 5:13-16 (ASV) 13 "Is any among you suffering? Let him pray. Is any cheerful? Let him sing praise. 14 Is any among you sick? Let him call for the elders of the church; and let them pray over him, anointing him with oil in the name of the Lord: 15 and the prayer of faith shall save him that is sick, and the Lord shall raise him up; and if he have committed sins, it shall be forgiven him. 16 Confess therefore your sins one to another, and pray one for another, that ye may be healed. The supplication of a righteous man availeth much in its working."
Notice that sin
is integrally connected to what Peter has written about being healed, the anointing with oil, etc.
So, I think there are two items that are being dealt with here:
1) Spiritual: suffering, both within and without, and rejoicing. In the former, prayer is to be exercised by the one afflicted through external circumstances (providence) and burdened internally perhaps due to sin, e.g., not willing to accept God's providence that has brought about situations which are not according to our personal desires
2) Physical: sickness of the body, which may be due to sin. There is certainly a connection here, don't you think? Whatever the cause of the sickness, it is obviously serious enough to warrant an overture for the elders of the congregation of which the person is a member to come and minister. The anointing with oil is often a means to relieve irritation and pain and in some cases to cure a malady, as was the practice in those days, especially given that physicians, e.g., Luke, were not plentiful.
Now, to me the fact that the elders are to be called must mean that there is a spiritual aspect to what ails the individual. This is the area of 'expertise' which elders are called to perform in regard to spiritual matters; to minister to the soul. And the "if he have committed sins, it shall be forgiven him
" statement seems to give credence to this also. While a Christian is absolved of all guilt through the blood of Christ, the temporal consequences of sin is sometimes not taken away but rather allowed to have its course, e.g., the death of David's son, the putting to death of someone who committed a capital crime, etc. Paul speaks about the putting out of an unrepentant sinner from the congregation in order that his 'flesh' may be destroyed, cf. 1Cor 5:5; 11:29,30.
To answer your specific question whether or not elders should be called when one is sick and that they might anoint the person with oil and pray over them... I think this practice is still valid for today when the circumstances warrant it. Okay, so what warrants the calling of the elders?
First, the anointing with oil is NOT a substitute for proper modern medial treatment. There are certain groups which believe this erroneous idea. Second, if the person believes that their illness is related to a sin committed. Again, the issue is two-fold; committed sins which have resulted in temporal consequences; in body and/or soul.