In the case that one academic institution publishes a journal but also they organize a conference each year in the same area. Would it be possible to publish the articles accepted at that conference in a special edition of this journal; instead of having separate conference proceedings? This journal is published quarterly. Is there any problem doing that or is somewhat atypical?
This set-up is quite common in Computer Science. The Association for Computing Machinery (one of the two dominant scholarly associations, the other one being IEEE) publishes a number of Reviews. The conference proceedings of a few stellar conferences are published as special editions of this otherwise "lightly reviewed" publication.
Second, many conferences make a deal with a journal for a special edition. The authors of the best papers at the conference are invited to submit an extended version of their paper to this special edition. An extended version is very briefly defined as having 30% additional material. A conference will often advertise this because it allows authors to claim a second publication in a more permanent form, important e.g. because some combined EE / CS departments insists that only journal articles count fully for promotion. The 30% rule also helps with the publisher of the conference proceedings who in general apply rights to the articles published. If your institute usually publishes the conference proceedings themselves (as opposed to e.g. IEEE Digital Library), there are no problems, provided the authors know about it beforehand.
As Buffy pointed out, you need to make sure that authors at submission know about your plans. In short, at least in my field, a special edition of a journal is reasonably common.
You need permission of all authors. You need permission of any copyright holders.
Usually it is the conference that arranges such things in advance and the conference submitters are fully aware of what will happen with their work. Doing it after the fact is much more problematic in terms of getting all necessary permissions.
If the authors have retained copyright then it is a logistical problem.