@jmcculley: I learned by watching how others tune while bowing. First tune the A, then A and D, then D and G, and last E. I hold the violin with my chin and shoulder (hands-free) and put the scroll between my 1st and 2nd fingers and hold the peg with my thumb and 2nd finger. The thumb is one one edge and the 2nd finger is on the opposite side and play the A as I loosen the peg about a half-step lower (about 1/4 inch) Next I play the A string as I tighten the peg a little bit until it is in tune. The thumb presses on one edge to loosen and pushes on the opposite edge to tighten. Push the peg into the peg box a little so it will stay in tune and not slip out. Your 1st finger will push on the scroll and your thumb and 2nd fingers will push in the opposite direction. I might have to go lose and tight a couple of times until I hit the pitch.
Next, play the D and A together. Grasp the D peg between the thumb and 1st finger again, but the peg is on the first knuckle of the 1st finger now. The 3rd and 4th fingers will press on the scroll on the opposite side. The left edge of the thumb is on the opposite side of the peg from the 1st finger. Loosen the peg the same as the A string and tighten it while pushing it into the peg box a little so it will hold a tune and not slip out. Your 3rd and 4th fingers will steady the scroll as your thumb and 1st fingers push the peg in. Move your thumb and 1st fingers so they push on opposite edges of the peg.
Next, I grasp the G string peg with my thumb and first finger. The scroll is between my 3rd and 4th fingers. For the G string, I grasp the peg with the tips of the thumb and 1st finger. The thumb pushes on one edge of the peg and the 1st finger pushes on the opposite edge of the peg and loosens it about a half step lower (about 1/4 inch). I play the G and D together to get the pitch right. You don’t want to touch the other pegs, so move your hands and fingers carefully around the pegs and scroll box.
Lastly, the E might be adjusted with the fine tuner. If you don’t have one, or if it slipped out of tune, the 2nd finger lies flat against the peg, the first finger presses on the opposite side of the scroll and the thumb switches from one edge to the other to tighten and loosen the peg.
Some pegs are easier to turn based on the direction they’re faced. After a while, I got comfortable re-positioning my hand and fingers to tune while bowing but it takes time and practice. Good luck!