The New Testament appears to describe at least 3 different avenues through which healing occurs. They are not mutually exclusive, and as noted by Dottard, in all cases the power comes from God.
Call for the elders
James encourages calling for specific individuals ordained to a specific office; he recommends an ordinance (anointing with oil). This squares with statements made by the Gospels, for example:
And when he had called unto him his twelve disciples, he gave them
power against unclean spirits, to cast them out, and to heal all
manner of sickness and all manner of disease (Matthew 10:1)
He that heareth you heareth me; and he that despiseth you despiseth
me; and he that despiseth me despiseth him that sent me (Luke 10:16)
22 And when he had said this, he breathed on them, and saith unto
them, Receive ye the Holy Ghost:
23 Whose soever sins ye remit, they are remitted unto them; and whose
soever sins ye retain, they are retained (John 20:22-23)
Jesus is giving people authority to act in His name. These authorized representatives go on to baptize, confer the Holy Ghost, heal the sick, and so on. In doing so they are acting not under their own power, but by authority delegated to them.
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The prayer of a righteous man availeth much
We are taught to ask God for the things we need & desire. One need not be ordained to a specific office in order to do this.
14 And when they were come to the multitude, there came to him a
certain man, kneeling down to him, and saying,
15 Lord, have mercy on my son: for he is lunatic, and sore vexed: for
ofttimes he falleth into the fire, and oft into the water.
16 And I brought him to thy disciples, and they could not cure him.
17 Then Jesus answered and said, O faithless and perverse generation,
how long shall I be with you? how long shall I suffer you? bring him
hither to me.
18 And Jesus rebuked the devil; and he departed out of him: and the
child was cured from that very hour.
19 Then came the disciples to Jesus apart, and said, Why could not we
cast him out?
20 And Jesus said unto them, Because of your unbelief: for verily I
say unto you, If ye have faith as a grain of mustard seed, ye shall
say unto this mountain, Remove hence to yonder place; and it shall
remove; and nothing shall be impossible unto you.
21 Howbeit this kind goeth not out but by prayer and fasting (Matthew
17:14-21)
In this story the father asks Deity for a miracle (he has the opportunity to do so in person; many others lacking this opportunity have asked in prayer). Jesus has the power & authority and personally heals the boy.
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The gift of healing
The passage from Matthew 17 cited above is instructive for this category as well. Jesus had given the aforementioned disciples power to act in His name, but in this case, either their faith or preparation or something else was inadequate to heal the boy.
This did not mean that the disciples could never expect to perform such a healing--but it does appear that some greater preparation & effort was expected of them in seeking "earnestly the best gifts" (see 1 Cor. 12:31).
Jesus had authority and He is the ultimate manifestation of One who has the gift to heal.
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Conclusion
- The faithful are taught to seek/ask from God blessings they could not
obtain on their own.
- Individuals are given divine gifts to enable them to fill different
functions in God's work--one of these gifts is the gift of being able
to heal.
- Some individuals are explicitly granted power to act as personal
representatives of Christ, and have power--from Him--to perform
healings (and other miracles) using His delegated power.
Paul speaks of one avenue for seeking miracles through God's power; James speaks of another.
Though a given individual may not have personal access to all 3 of these avenues for seeking God's power, there is no reason why a single person cannot have all 3. All 3 make an appearance in the passage cited in Matthew 17.
