I used the Horizons data from NASA. You can find it here.

I pulled 7500 days of data, and calculated the distance at each time between Earth and Mercury, Earth and Venus, and Earth and Mars.

I made a plot summarizing the results:

The color bars at the bottom are the color of the closest planet at that time (blue for Mercury, yellow for Venus, and red for Mars).

My hunch on Mercury appears correct, but it's close between Mercury and Venus. One of these planets is always closest since the largest gap between Mercury and the Earth is smaller than the smallest gap between Earth and Jupiter.

This might be a bit suprising to some because the distance from Venus to the Sun is closest to the distance from Earth to the Sun, and Venus does come closer to the Earth than any other planet. If you trace out orbits though, you'll see that that doesn't guarantee the smallest average distance because Venus can end up on the other side of the Sun for an extended period compared with Mercury.

I'll try to put together a gif showing this at some point.