Is requesting a supplication from a shaykh considered an endorsement of him, is it permissible for a Muslim to ask shaykhs for supplication, and what are the conditions for that?

It is permissible to ask others to supplicate on one's behalf, provided that this does not become a habit leading to negligence in supplicating for oneself. Also, one should be mindful not to cause the person being asked to supplicate to become conceited or self-admiring, and the intention behind the request should be to benefit both oneself and the person supplicating. Merely asking for supplication does not guarantee its acceptance, and what is forbidden is to make definitive declarations of a person's piety, as the Prophet, peace and blessings be upon him, said: "I reckon so-and-so, and Allah is his reckoner, and I do not sanctify anyone before Allah." The important thing is that one's concern should be directed towards Allah, who answers the distressed.