First off, the notation. For each move, the first letter would be the priest label, then < or > for left or right, then a number denoting the number of steps. If a priest move (up or down) on the same tile, I'll just specify the priest label by itself.

39 Moves:

1. C<2
2. C<3
3. C<5
4. C<1
5. C
6. C>2
7. C>1
8. C<4
9. C
10. A>1
11. A
12. C
13. B>2
14. B>1
15. B
16. A>2
17. A
18. A>3
19. A
20. C
21. C>1
22. C
23. C>2
24. C>1
25. C>4
26. C
27. A
28. A>2
29. B>5
30. A
31. B
32. C
33. D<1
34. C<3
35. C
36. D<3
37. A<4
38. B
39. B<5

The idea is:

to first notice that only C can move, and C has two options: C<2 or C<3.
To enable A, we need one priest on the first blank tile next to A, so that's our first goal (move 1-9).
To enable B, we need two priests on B's row, so B can do B>2. This can simply be done by A and C (move 10-12)
To enable D, we need three priests on D's row, so D can do D<1. This means all three priests must not land on the peak, but must cross first to the right side and enable D (move 13-32), before moving on to the peak.
Afterwards, it's straightforward to move them into the peak (move 33-39)

First off, the notation. For each move, the first letter would be the priest label, then < or > for left or right, then a number denoting the number of steps. If a priest move (up or down) on the same tile, I'll just specify the priest label by itself.

39 36 Moves (update thanks to bobble, I switched A and C to save 3 moves):

1. C<2
2. C<3
3. C<5
4. C<1
5. C
6. C>2
7. C>1
8. C<4
9. C
10. A>1
11. A
12. C
13. B>2
14. B>1
15. B
16. C>2
17. C
18. C>3
19. C
20. A>2
21. A>1
22. A>4
23. A
24. C
25. C>2
26. B>5
27. A
28. B
29. C
30. D<1
31. A<3
32. A
33. D<3
34. C<4
35. B
36. B<5

The idea is:

to first notice that only C can move, and C has two options: C<2 or C<3.
To enable A, we need one priest on the first blank tile next to A, so that's our first goal (move 1-9).
To enable B, we need two priests on B's row, so B can do B>2. This can simply be done by A and C (move 10-12)
To enable D, we need three priests on D's row, so D can do D<1. This means all three priests must not land on the peak, but must cross first to the right side and enable D (move 13-29), before moving on to the peak.
Afterwards, it's straightforward to move them into the peak (move 30-36)