I tried to recreate this issue on a fresh installation of vBulletin 5.1.5 with no modifications. Using the [html] tags I get this result:
Using no BBCODE, I do get issues so that will need to be checked and tested more but it doesn't break the entire site. Here is what is does if no BBCode is used:
In both occurrences, the Can Use HTML permission is off globally for Administrators.
Though if you've made other changes to the templates and the above code is compounded on errors that browsers could work around, then this could cause more issues.