Either is possible, but their grammar is different.
May he succeed.
Let him succeed.
May is a modal, and takes a verb (in the base form), not a clause. This is an inversion of He may succeed, used for wishes. Other than with may, the syntax is now obsolete except for a few set expressions like Long live X!
Let is not a modal, but a verb which takes an infinitive clause. It normally has as a subject the agency allowing the action, so Let him succeed is syntactically a command, and could be said to somebody who has the power to hinder the success. But it is also used in a more general way, and I think is now more common than the "may" construction for that purpose.