Some of the introductory classes at WashU are going to be large. CSE 131 (the introductory programming class) is around 500-700 students. It’s a flipped class though and the only time you attend any classes are for the labs/midterms.
Upper division classes will be smaller obviously,.
I wouldn’t worry about not getting classes. Even though entering freshman register last, they will get into their classses without much problems. Which is the advantage of a private school as opposed to a public school.