There are a lot of schools that are large in enrollment, that may be powerhouses in some NCAA sports, but lacking an NCAA hockey team... however, many of them have club hockey governed by the American Collegiate Hockey Association (ACHA). The players generally have to pay to play and do not get athletic scholarships (there is some funding available occasionally), but despite this, the level of play can be pretty high. There are over 400 men's ACHA teams (split into three divisions) and about 75 women's teams (split into two divisions). I did not see Auburn on the list of current women's teams, but they could have of course had a women's ACHA team in the past.
I have personal experience with two of them (well, the second one is simply knowledge that it exists). Arizona State University's hockey program began as a "club" (i.e. not sponsored by the NCAA) team and eventually joined the ACHA at some point, probably as it formed, but I won't bore you with the details and full history by looking it up. ASU is located in my hometown and where I went to college, and by the time I started going to ASU (2004), the men's hockey team was an ACHA powerhouse, and eventually won an ACHA D1 championship in 2014. The women's team also was a D1 runner-up in 2001. Through this and its growing popularity, a big donor came in and helped pave the way for the men's team to make the jump up to being an NCAA D1 program, starting with a split ACHA/NCAA schedule in 2015-16 and then a full NCAA schedule from 2016-17 onwards. They originally played at Oceanside Ice Arena in Tempe (which is where I learned to skate as a kid, and also eventually where I started learning to play hockey), and that big donation (to the tune of $32 million) helped fund Mullett Arena, which is where the play now, and also that farcical situation where the Coyotes played there for two seasons.
Meanwhile, now that I live in Gainesville, the local university (University of Florida) has a club team, and looking it up, they're members of the Amateur Athletic Union rather than the ACHA. They won the AAU D2 national championship last year. I know very little about them other than the fact that they exist. Gainesville has no ice rinks, and although the nearest ice rink is over in Jacksonville, UF Gators hockey practices and plays at a rink that's down in the northern part of the Tampa Bay area, even though it's a bit further away than going to Jacksonville for ice time. I'm not sure why, but there's probably just some kind of scheduling conflict with limited availability for ice time. I guess it's not too big of a deal to travel a bit extra for ice time if you're already going to have to make a trip of well over an hour (it's about an hour and a half drive to Jacksonville, and a little over two hours to get to the northern part of the Tampa Bay area).
