A recent Harvard Law School class had 562 students that were drawn from 188 different undergraduate colleges.
The undergrad brand name has surprisingly little to do with a kid’s ability to get into a top law school. You need a strong high GPA transcript and a high LSAT score. Kids who can get into top 15 schools like Vandy will tend to be good at doing those two things. So attending a top 15 school is correlated to doing well in law school admissions, but is not really causing the kids to do well in law school admissions. Those kids would do well wherever they attended college.
My 3.9998 GPA from a meh undergrad college was perfectly fine to get me into a top 10 law school. We had one or two kids a year get into HLS. While I had plenty of law school classmates who attended fancy top undergrad colleges, I also had plenty of classmates like me.
In fact, you could argue that it was easier for me to put up strong GPA numbers at my meh college than it would have been at a top 15 college. Given the expense of law school these days (and the relatively meh job prospects for newly minted J.D.s), you can also argue that it is dumb to blow the 529 on just to obtain a name-brand undergrad degree.
So law school admissions odds is perhaps the single worst reason to pick an undergrad college. Your kid should pick someplace that he likes, that he can get admitted to, that is a fit for his interests, that you can pay for, which has good food in the cafeteria, and that has cool school colors and mascot. Law school will take care of itself (or not) regardless of where he goes.