The Buddha is the greatest among all doctors,
able to cure all diseases of afflictions,
able to relieve people from all the great sufferings of birth and death.
— Avatamsaka Sutra
Yes, Buddhists firmly believe in the efficacy of prayer. When someone prays, his faith engenders a mental state of supernormal, unified concentration, by which he can stimulate or arouse (gan) the compassionate vow-energy of the beings (such as Buddhas or Bodhisattvas) to whom he prays, and thereby receive a response. That is, the mental energy resulting from the supplicant’s concentration tallies and interacts with the energy of a Buddha’s or Bodhisattva’s vows. This interaction, in turn, gives rise to an inconceivable extraordinary power, which produces the special experiences and efficacious results of prayer.
In Chinese societies, where Mahayana practice prevails, praying to the Bodhisattva Guanyin (Sanskrit: Avalokiteshvara) [for example], brings about… miraculous results. However, a benevolent deity or spirit actually already protects a reverent devotee of the Three Jewels [Buddha, Dharma, Sangha]. If she suddenly encounters misfortune, even if she doesn’t pray at that very moment, her ill ‘luck’ [karma] will be averted, provided that she has firm faith. Prayer works because of firm conviction. If someone has unyielding faith, she is constantly under the protection of prayers.
Although Buddhists believe in the power of prayer, they do not emphasize that praying can solve every problem. For example, if a Buddhist becomes ill, prayer is very important, and if she has confidence, then in certain cases, through pious, sincere prayers she can induce a sympathetic response and be cured. Therefore, we find Buddhists scriptures mentioning cases of people being cured through expounding or hearing the Dharma. But if the sick person lacks conviction and experience in (successful results from) prayer, she should se a physician. So even during the time of the Sakyamuni Buddha, when a bhiksu became sick, he was normally treated with medicine. Therefore, it is generally believed that the Dharma is primarily for treating the spiritual illness of birth and death. For physical illness, one should use medicine. The ill should pray but also seek medical treatment: this is the view of an orthodox Buddhist.
Orthodox Chinese Buddhism
Chan Master Sheng Yen