Yes, Mormons believe in gifts and miracles of both spiritual and physical healing in these days.
Elder Dallin H. Oaks of the Quorum of the Twelve Apostles gave a talk in 2010 called Healing the Sick. The whole talk is relevant to your question but here's a snippet (not sorry for making it essentially my whole answer):
Latter-day Saints believe in applying the best available scientific knowledge and techniques. We use nutrition, exercise, and other practices to preserve health, and we enlist the help of healing practitioners, such as physicians and surgeons, to restore health.
The use of medical science is not at odds with our prayers of faith and our reliance on priesthood blessings. When a person requested a priesthood blessing, Brigham Young would ask, “Have you used any remedies?” To those who said no because “we wish the Elders to lay hands upon us, and we have faith that we shall be healed,” President Young replied: “That is very inconsistent according to my faith. If we are sick, and ask the Lord to heal us, and to do all for us that is necessary to be done, according to my understanding of the Gospel of salvation, I might as well ask the Lord to cause my wheat and corn to grow, without my plowing the ground and casting in the seed. It appears consistent to me to apply every remedy that comes within the range of my knowledge, and [then] to ask my Father in Heaven … to sanctify that application to the healing of my body.”
Of course we don’t wait until all other methods are exhausted before we pray in faith or give priesthood blessings for healing. In emergencies, prayers and blessings come first. Most often we pursue all efforts simultaneously. This follows the scriptural teachings that we should “pray always” (D&C 90:24) and that all things should be done in wisdom and order.
Truly, as the Book of Mormon teaches, God “manifesteth himself unto all those who believe in him, by the power of the Holy Ghost; yea, unto every nation, kindred, tongue, and people, working mighty miracles … among the children of men according to their faith” (2 Nephi 26:13).
Many scriptures teach that the servants of the Lord “shall lay hands on the sick, and they shall recover” (Mark 16:18). Miracles happen when the authority of the priesthood is used to bless the sick. I have experienced these miracles. As a boy and as a man I have seen healings as miraculous as any recorded in the scriptures, and so have many of you.
You might say that the LDS believe in modern medicine as a miracle in and of itself. Elder Russell M. Nelson, another member of the Twelve, is a world-renowned heart surgeon who developed several new techniques for saving lives. But, as he has stated in his own talks on the matter, not without faith and prayers and fasting.
The Book of Mormon says, in Moroni 7:27,29,35-36:
27 Wherefore, my beloved brethren, have miracles ceased because Christ hath ascended into heaven, and hath sat down on the right hand of God, to claim of the Father his rights of mercy which he hath upon the children of men?
29 And because he hath done this, my beloved brethren, have miracles ceased? Behold I say unto you, Nay; neither have angels ceased to minister unto the children of men.
35 And now, my beloved brethren, if this be the case that these things are true which I have spoken unto you, and God will show unto you, with power and great glory at the last day, that they are true, and if they are true has the day of miracles ceased?
36 Or have angels ceased to appear unto the children of men? Or has he withheld the power of the Holy Ghost from them? Or will he, so long as time shall last, or the earth shall stand, or there shall be one man upon the face thereof to be saved?
37 Behold I say unto you, Nay; for it is by faith that miracles are wrought; and it is by faith that angels appear and minister unto men; wherefore, if these things have ceased wo be unto the children of men, for it is because of unbelief, and all is vain.