The chords are quite simple, no matter which of the two distinct versions you're looking for. Transpose to your favorite key.

Rodgers sang it as a 3-line blues, with the third line yodeled:

[C]Good morning, Captain, Good morning, Shine
[C7] Do you [F] need another mule skinner out on your [C] new mud line?
Yodel-ay-hee [G7]hee, yodel ay hee ay hee [C] hee.

The more common post-Jimmie versions do it as a 3-line blues, but set up like this, with the first line repeated as in standard blues format:

[C]Good morning, Captain, Good morning, Shine
Good [F] morning, Captain, Good morning [C] Shine
Do you [G7] need another mule skinner, out on your [C] new mud line.

In this version, the yodel (if it's used at all) comes after the third line and uses the same structure as the yodel in the first version. For some variety, an extra yodel can be thrown in between verses, on the order of:

[C7]Yodel ay-[F] hee hee hee
Yodel [C] ay-hee-hee hee
Yodel[G7] ay-hee ay-hee oh-lay-[C] ee.