Is it inappropriate to tell my students to check their school email?
At every (US) institution I've attended or worked at, there's been an official school-wide policy that students are expected to check their school email regularly, and that sending official communications there constitutes sufficient notification.  So check your school's policies; if you have such a policy, then it's certainly appropriate to remind students about it.

Moreover, in such a case, you can enforce it: if you send an email to school accounts saying "Please respond by date X if you want Y", and they don't respond by date X (whether because they didn't check their email or for any other reason), you may not be under any obligation to give them Y.  (Disability accommodations could have special rules so contact your school's disability office to check on that.)