Having visited Chaco many times in the past 20+ years the dusty 13-mile dirt road accessing the canyon from the US550 can be a challenge when pulling an Airstream. However, go slow and just get there.
Recommend planning for a 2 or 3-night stay to allow plenty of time for walking/hiking around to some of the more remote sites. Most people that visit drive there for a "day-trip" - very few folks take the easy hikes to the interesting sites that are off the beaten path.
My last visit there was a motorcycle ride in October and it was fantastic. Great weather - an elk herd even met us at the entrance! There are 20+ primitive camp sites for trailers and more than double that if your interested in tent-camping only. The camp host was an Airstreamer!