Richard Gaffin, who provided the cessationist perspective in Are Miraculous Gifts for Today?, says in his book Perspectives on Pentecost (pg. 114) that individual endowments of healing were "foundational gifts" which have "passed out of the life of the church." But he says more:
At the same time, however, the sovereign will and power of God today to heal the sick, particularly in response to prayer (see, e.g., James 5:14, 15), ought to be acknowledged and insisted on. ... Healing ... ought to be an expectation of God's people today.
This seems to be agreed by most cessationists. But beyond this, it seems that the interpretation of James 5 is quite disparate between individual cessationists. There's not a whole lot of agreement other than the point that, "God still heals people today."
In Saved by Grace (pg. 38), Anthony Hoekema (an influential CRC pastor and reformed systematic theologian) says:
The main theme of the passage is not the efficacy of anointing with oil but the power of prayer--see the concluding sentence and note the example of Elijah in verses 17-18. The situation James here describes is that of a person who is obviously "too sick to go to church"; hence he is invited to ask the elders to come to his house. ... The grammatical structure of the sentence suggests the possibility that the elders first rubbed the sick person with oil (presumably olive oil) and then prayed over him (or her). The application of the oil, however, was for medicinal purposes. James is saying, in other words, that the church’s ministry to the sick should include the best medical help that can be found.
B. B. Warfield would seem to agree with Hoekema. In The Theology of B. B. Warfield, Fred Zaspel summarizes pages 169-173 of Warfield's Counterfeit Miracles:
"The emphasis falls wholly on the sick man's getting himself prayed for officially by the elders of the church, and the promise is suspended wholly on their prayer, on the supposition that it is offered in faith." And the anointing with oil Warfield takes in a medicinal sense. Oil was a universal medicine in the day, and the word for anointing is not the ceremonial term but the term meaning "to rub." The exhortation is to offer medical help while looking to the Lord for healing. But even if the oil is symbolic of the Holy Spirit and the anointing a ceremonial rite, there is no exclusion here of the ordinary medical attention. More to the point, there is nothing of miraculous gifts. God heals in answer to prayer, and that is the promise this passage makes.
Herman Hanko, a PRC pastor and editor of the Protestant Reformed Theological Journal, wrote in the May 1975 edition of that publication:
Although it is true that the word which the apostle uses here is oftentimes used in Scripture for physical sickness, this is by no means always the case. ... This word and its cognates is therefore often used in Scripture to refer to spiritual weakness. See such passages as I Cor. 2:3; Heb. 5:2; 7:28; 4:15; II Cor. 7:5-9. ... The context refers very emphatically to sin: "and if he have committed sins, they shall be forgiven him." ...
The text speaks emphatically and without qualification of this prayer being answered with healing. But it must be remembered that we have in all the Scriptures no promise of God that He will always make us better when we are physically sick. Not even faith healers dare to claim this absolutely. We conclude therefore, that the reference of the text is to spiritual weakness. ... Inability to pray is the spiritual sickness or weakness to which the apostle refers. Such spiritual weakness may come for various reasons. ... In a very real sense, when the saints call the elders they call Christ Himself. ...
The apostle adds, "anointing him with oil." This phrase must not be taken in the literal sense of the word. James uses many such Old Testament figures which he does not intend to be taken literally, but which must be taken in the spiritual sense. For example, in chapter 4:8 the apostle refers to Old Dispensational forms of cleansing when he says: "Draw nigh to God, and He will draw nigh to you. Cleanse your hands, ye sinners; and purify your hearts, ye doubleminded." Rather, oil is a symbol of the Holy Spirit. And anointing with oil in the Name of the Lord means that the prayer which is offered as an intercessory prayer by the elders is a prayer for the Holy Spirit. It is a prayer that the Spirit may come upon the sick and restore those sick to spiritual health.
His fellow PRC minister Rodney Kleyn agrees.
John MacArthur takes a very similar approach:
The weak are those who have been defeated in the spiritual battle, who have lost the ability to endure their suffering. ... Metaphorically, the elders’ anointing of weak, defeated believers with oil conveys the responsibility for elders to stimulate, encourage, strengthen, and refresh (cf. Luke 7:46) these people.