(Each chord is half a measure in 4/4)
A: GGCC GGDD GGCC DDGG
B: EmEmCC GGDD EmEmCC DDGG
However, in a fiddle contest setting, the backup for this tune is typically done in a Texas style backup. In this style, there is a walking bass line beneath the chords, and frequently the chord will change every half-measure.
The chord progression using Texas Backup:
A: G6 G7 C C#o G6 G7 D7 D7/A G6 G7 C C#o D7 D7/A D7/G# G6
B: Em G C C#o G6 G7 D7 D7/A Em G C C#o D7 D7/A D7/G# G6
I'm including chord diagrams so you can see the bass line walk. Note that the X in the chord charts is the bass note that will be picked. Then the rest of the notes will be strummed on the next beat. (boom-CHUCK, boom-CHUCK, boom-CHUCK, boom-CHUCK).
You'll probably want to use the chord charts attached to understand what's going on better. It does take some work to learn some of these new chord shapes, but I think it makes playing backup a lot more fun. Try it out!
Also note that part A is a pretty standard G progression, so there will be other songs that will use this same progression. Also, since most of the chords are closed chords (no open strings), the same progression can be played with little additional effort by just starting two frets higher on an A6 instead of a G6.