According to LDS teaching, did Job have the priesthood?
It would make sense for Job to be a high priest, since his life pointed towards Christ (which is a main duty of high priests), but I could find any sources talking about Job's priesthood.
Job lived during a time when a father was the priest for his family. Job 1:5 says,
When a period of feasting had run its course, Job would make arrangements for them to be purified. Early in the morning he would sacrifice a burnt offering for each of them, thinking, “Perhaps my children have sinned and cursed God in their hearts.” This was Job’s regular custom.
At the end of the book, in Job 42, it says:
7 After the Lord had said these things to Job, he said to Eliphaz the Temanite, “I am angry with you and your two friends, because you have not spoken the truth about me, as my servant Job has. 8 So now take seven bulls and seven rams and go to my servant Job and sacrifice a burnt offering for yourselves. My servant Job will pray for you, and I will accept his prayer and not deal with you according to your folly. You have not spoken the truth about me, as my servant Job has.” 9 So Eliphaz the Temanite, Bildad the Shuhite and Zophar the Naamathite did what the Lord told them; and the Lord accepted Job’s prayer.
10 After Job had prayed for his friends, the Lord restored his fortunes and gave him twice as much as he had before.
In this instance, Job is called upopn by God to act as a priest to accept sacrifices on behalf of his "friends". However, no mention is made of creating an enduring, hereditary priesthood as was the case with Aaron and Levi.
So I would say that Job was a priest.