Is it common to not send rejection letters? I have been applying to some summer programs for undergraduates recently and I am seeing that some of them do not send rejection letters or emails, and instead only notify accepted applicants. I was wondering how wide spread of a practice this is in academia, meaning do graduate schools do this, do grants do this, et cetera?
Based on rumor, it is quite common not to send rejection letters at American universities.  I would expect that this happens at a substantially higher rate for undergraduate summer research programs because they have an exceptionally high ratio of applications to funding.  At many universities, the summer research programs are more competitive than the PhD programs but have zero administrative staff.  PhD programs, postdoctoral positions, and faculty positions also may not send rejections.  For some jobs, one will also find that nobody was hired at all.

Some Australian universities have an official no rejection letter policy.

I definitely recommend sending rejection letters, particularly if applicants paid an application fee.