Bottom line: Yes, but you’ll need specialized equipment and skills to do it.
I know people who have skied through the Enchantments as an overnight backcountry adventure. It’s a different and beautiful landscape in snow. You don’t have to follow trails, you can camp wherever you like, and there aren’t any permit restrictions.
Don’t expect a winter trip to be anything like summer. Here are some of the factors to be prepared for:
Snow. Several feet of it, and as much as 15 feet in wind-deposited areas. There’s not an established snow trail in winter, so you’ll need something on your feet for flotation, like backcountry skis or good snowshoes.
Darkness. The days are short in winter. Up here, dark really is dark and the temperatures plummet. You’ll spend a lot of hours in your sleeping bag trying to stay warm.
Cold. Don’t expect the temperatures to be above freezing at any time during the winter quarter. Past weather is no indication of future luck, but on Weather Underground you can search back year by year and see past weather in Wenatchee, elevation 800′. The Core elevation is 6200-7200′, so, about 30°F (17°C) cooler than this.
Steep trails. They’re no steeper in winter than summer, but they’re covered in snow and ice. You may need some technical climbing gear, and the mountaineering skills to go with it, to get safely up and down, especially on the Snow Creek Wall.
Avalanches. The route into and out of the Core is avalanche terrain. The trail through the Core itself mostly avoids avalanche terrain and runouts. However, you won’t see any sign of the trail. Use an online tool like Gaia GPS mapping software to evaluate slope angles. Combined with your AIARE or A3 training, you can find a safe time and route for your trip.
Weather. It’s a wildcard in the alpine year round. Mountain weather has a nasty temper. Bad weather in winter can be dangerous. You’ll need to be prepared for anything.
That said, a winter traverse of the Enchantments is a trip that someone does every winter. Being prepared can make the journey a memory of a lifetime. Being unprepared can cost you your life.
Waiting until spring can mitigate some of the negative factors above. The days are longer, the weather is warmer, the snowpack is often more stable. This photo was taken in spring and is typical of the Core in May.
Take time this winter to launch your own hiking web site. Start with this great domain.