It is important to distinguish between cessation of spiritual gifts being bestowed on individuals, and cessation of all miraculous events entirely. Cessationists consider the first case (individual bestowment of spiritual gifts) to have ceased, but not all miraculous events entirely.
See B. B. Warfield's Counterfeit Miracles, in the section titled Faith Healing:
"In the healing of every disease of whatever kind," writes Doctor
Henry E. Goddard, "we cannot be too deeply impressed with the Lord's
part of the work. He is the operator. We are the co-operators. More
and more am I impressed that every patient of mine who has ever risen
up from his sick bed onto his feet again has done so by the divine
power. Not I, but the Lord, has cured him. And it is this fact that
the Lord does so much, that gives to different systems of healing
their apparent cures. He has healed many a one in spite of medicine,
in spite of mental healers, in spite of ignorance, in spite of
negligence and poor and scanty food. Nineteen out of twenty cases of
grippe will get well without doing anything for it, if we are willing
to bear it until that time. Pneumonia, even, is what the physician
calls a self-limiting disease, and many cases will recover alone if we
are willing to run our chances with it. The arm may drop into boiling
water and become scalded. Nine times out of ten it will take care of
itself and heal. But if that arm is mine it is going to have an
outward application which will make it feel better the moment it
touches it. And more important by far, it is going to be dressed
aseptically to prevent blood poisoning. It might get well itself,
probably would; but it is going to have my little co-operation, the
most intelligent that I can render, that the Lord may have the open
door through which He can come in and bless it." It is the very spirit
of James, I take it, that speaks in this Christian physician. If you
are sick, you will use means, all the means that exist; but you will
use the means in the name of the Lord, and to Him you will look for
the issue. (172–173)
In short, cessationists hold that the sovereignty of God is such that all healing of any sickness is the result of God's working, and also hold to the miracle of salvation as the pre-eminent form of still-continuing miracle. Indeed, the miracles that have ceased are those apostolic-era gifts used only for the purpose of spreading the gospel in the early church; those that were bestowed on individuals, rather than purely by God's working.
In James 5:14, no individual has been bestowed with a gift of healing. Rather, with prayer foremost, and secondarily medical care in the form of anointing of oil (historically considered to have medicinal value), a group of the elders of the church were to care for the unwell individual. The hope is that God would then miraculously heal the individual. However, this is quite a different circumstance to that of the apostles and those granted the gift of healing, who could "at will" heal anyone they chose.
With that in mind, cessationists would encourage this practice (perhaps not with oil specifically, but the best available medical care), but would not consider any of the elders to have been bestowed any spiritual gift of healing in the event of a miraculous healing.
It is worth noting that John Calvin, in his commentary on James 5, considered the anointing with oil, prayer, and resulting healing to be an apostolic sacrament which has passed. This is no longer the predominant position among cessationists, to the best of my knowledge. However, even at the time he wrote, others considered the anointing with oil to be medical (a common view today as well), and thus only the prayer was efficacious, in as much as God chose to heal the sick in that instance.
