Jehops' Description of Work
Previously, the shell-integration scripts (emacs-vterm-bash.sh, emacs-vterm-zsh.sh, emacs-vterm.fish) were installed under ${EMACS_SITE_LISPDIR}/etc/, i.e. share/emacs/site-lisp/etc/. That is a shared, generically-named directory, and creating it cause problems for at least one other package [0].
Install everything under ${EMACS_SITE_LISPDIR}/vterm/ instead. This keeps the port self-contained, no longer pollutes a directory other ports probe, and matches ELPA package's layout.
[0]
math/ess auto-detects ess-etc-directory by probing paths relative to its own lisp directory, one of which is ../etc. With site-lisp/etc present, ESS selected it instead of the correct share/emacs/etc/ess, so M-x R failed to load ESSR with "cannot open the connection".