The short answer is 'no' ... but it's really a qualified no. Here's why:
Position of the Milky Way band
The Milky Way is a band that appears to go all the way around the planet. Roughly described, is the plane of the Milky Way galaxy. In January the night-side of the planet sees the side of the Milky Way facing away from the galactic center -- so the Milky Way band is thinner. In July the night-side of the planet sees the side of the Milky Way facing toward the galactic center.
But since the band of the Milky Way appears to rise (really the planet is rotating) ... the time during the night when the Milky Way appears to be most prominent will vary based on which month or day you plan to do your imaging.
Currently (this being early June), the Milky Way is barely rising after dark in the evening hours. If you want the Milky Way positioned vertically ... you'd be waiting until ... maybe 2-4am. So this brings us back to the Moon.
The takeaway here: The best time to shoot the Milky Way during the night will change depending on the desired placement (e.g. Do you want a vertical Milky Way vs. do you want a Milky Way with a diagonal path or one that arcs across the sky? etc.) The best time to shoot during the night to get your desired placement will vary based on the date.
While a New Moon is rising and setting with the Sun. The Moon isn't visible at any time during the night. This gives you maximum flexibility.
A First Quarter Moon rises roughly in the middle of the daytime hours and sets roughly in the middle of the nighttime hours. This means if your best time to shoot the Milky Way is in the pre-dawn hours ... the Moon would already have set.
A Last Quarter Moon rises roughly in the middle of the night and sets roughly in the middle of the day. This means if your desired time to image the Milky Way is after sunset (the early part of the night) then the Moon will not have risen in the sky yet.
And of course the Full Moon is up all night long ... there is no good time during the night to shoot during a full moon.
Check the times for moon rise & set. Each night. The Moon will rise, transit, and set about 55 minutes earlier each day.
The primary reason to avoid shooting when the Moon is up is because of the large amount of light pollution created by the Moon. It's the same reason to travel away from cities -- to get away from the street lights.
Light from the Moon (or any light source) is illuminating particles in the air... dust, smog, humidity, etc. and this means the "black" background of space isn't so black anymore... it's more of a dark gray. That gray starts to become brighter than the fainter stars in the sky (or any other deep sky objects) and makes them more difficult to see... so the image is less dramatic.
Can you have a tiny bit of light pollution? Sure... it's just less optimal and this means you'll need to "stretch" your image data a bit more to try to tease out the Milky Way in your final images. When shooting the Milky Way, you're always going to be heavily post-processing the data to "stretch" it to improve the visibility.
BTW, Astronomical Twilight ends when the Sun is 18° (or more) below the horizon. After that point, the atmosphere is unable to refract any of the Sun's light farther around the planet -- any light you still see once the Sun is beyond 18° below the horizon is coming from something other than the Sun.
There are a number of applications that can make it easy to work out the position of the Milky Way as well as the Moon or Sun.
Stellarium is a free desktop planetarium application that runs on Windows, Mac, & Linux machines ... you can't beat the price. There are also a number of non-free apps.
There are also some mobile apps that are nice. I use Sky Safari on my phone (extremely helpful for general astronomy as well). But there is a photography-specific app that I've found called "PhotoPills". The most interesting (to me anyway) feature of PhotoPills is that you can scout locations to shoot during the daytime and use it's augmented-reality feature -- the phone will display the image of your scene. You set the date/time when you'd like to shoot and it will overlay the night sky and Milky Way band so you can evaluate how the Milky Way will appear relative to other elements of your scene ... to work out the best location to set up for the shot. Return back at the time and location to capture the Milky Way exactly as you want it. PhotoPills used to be iOS only, but is now also available on Android.