Yes, during dormancy is almost always the best time to transplant deciduous plants. If your crape myrtle was planted within the last year or two, you could probably get away with moving it now, as it won't have extended its root system very far. If it is well established, though, it will have a much better chance of surviving when dormant.

Wet, muddy soil will be mostly an inconvenience to you, not the plant. Just try not to compact the soil around the new location. I suggest getting a piece of wood or plywood to stand on to distribute your weight. Using only the native soil without amendments is fine, in fact that is recommended for woody plants.

One suggestion would be to loosen up the new site while the soil is not too soggy, then cover it with a tarp to keep the rain off it. This might make it a little easier to plant in next spring.

Also, crape myrtles are very late to leaf out in our climate, so you may find that things are drying up next spring before it starts to leaf out. It is fine to transplant it just as new growth starts.

If the site is truly waterlogged - with standing water for extended periods - it is not a good site for a crape myrtle, which need decent drainage in winter.