Assume the pegs are in the upper left corner of a slightly enlarged chess board, which has indices $1 - 9$ and A - I. Now make the moves
b8-d6, c7-e5, d6-f4, e5-g3, f4-h2, g3-i1
b9-d7, c9-c7, c8-e6, c7-e7, d7-f5, e7-e5, e6-g4, e5-c5, f5-h3, g5-g3, g4-i2, g3-i3
a7-c7, a9-a7, a8-c6, c7-c5, a7-c7, b7-d5, c5-e5, c7-c5, c6-e4, e5-e3, c5-e5, d5-f3, e3-g3, e5-e3, e4-g2, g3-g1, e3-g3, f3-h1
If I'm counting right, the wonderfully animated solution of @squeamish ossifrage has $12 \times 3 = 36$ moves, which is the same number of moves as my solution above. Inspired by the animated solution, I found that I can move the $9$ peg block two places down and to the right with just $9$ moves:
b8-d6, c7-e5, b9-d7, c9-c7-e7-c5, a7-c7-e7, a9-a7-c7, a8-c6-e6, c8-c6, b7-d5
This reduces the total number of moves to $9 \times 3 = 27$ moves. I don't know if this is the minimum, but it's a start.
Made a computer program to look for a solution with fewer moves. It managed to improve my previous solution so it now only takes
Here they are:
a8-c6, b9-d7, b7-d5, c7-e5, a9-c5, c9-e7, a7-c7, b8-f4, c8-g4, c6-e6, e5-e3, c5-e5, d5-h3, g4-i2, e5-i1, e7-i3, c7-g3, d7-f5, e6-g2, g3-g1, e3-g3, f5-h1, f4-h2
I had to make some assumptions to get a solution within a reasonable time, so I'm not absolutely sure this is the minimum. I'd love to see the minimum if this is not it!