This issue was discussed in detail by ShaykhulIslam Abu Al'Abbas ibn Taymiyyah (may Allah be merciful to him) in his many useful books, such as "AlQa'idah AlJalilah fil Tawassul wal Wasilah," which is a useful book worth reading.
According to Ijma' (consensus of scholars), the same ruling of permissibility applies to living persons other than the Prophet (peace be upon him). For example, it is permissible to say to your brother, father, or any righteous person, "Supplicate to Allah to cure me of my illness, to restore my eyesight, to provide me with righteous offspring, etc." Allah is the One Who grants success!