Is it common to not send rejection letters?
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.