Some exoplanets are found by measuring how their star wobble. I was wondering how much the Earth wobbles in orbit due to the Moon's gravity, how far it actually would move.

The Earth does wobble on its orbit due to the Moon's gravity. Below is a representation of Earth's trajectory around the Sun where the wobble has been amplified thousandfold.

Both the Earth and the Moon orbit around the Earth-Moon center of mass, but the Earth has a much smaller orbit because it is much heavier. The size of the wobble can be computed from the Earth-Moon distance multiplied by the mass ratio, which gives around 5,000 km (slightly less than the Earth's radius, 0.003% of the Earth's distance to the Sun). The timescale for one wobble is the same as the orbital period of the Moon, 27 days. It corresponds to a relative velocity of 0.01 km/s, which is small compared to the orbital velocity of the Earth around the Sun, which is about 30 km/s.