Do reputable journals ever conduct peer review entirely by email?

I have interacted with several reputable math/TCS journals, both as author and as reviewer, entirely via email without the involvement of an online submission. I have even witnessed senior members of the community expressing the sentiment that they never agree to the semi-automated referee requests sent by online submission systems, but only to referee requests sent "by hand".

Summary: Yes, there exist reputable journals conducting peer review entirely by email.

Now whether or not a journal soliciting peer review without an online submission system is more likely to be disreputable than one that does, I do not know. I see no particular reason to assume either way.

Not long ago all reviews were handled not by email, but by paper mail. I submitted my first paper in 2003 and at that time submission was still by paper. However, I got an acknowledgement that the paper has been received via email, but the response of acceptance from the editor was in paper again. An online system was not in sight at that point.

I would say that there is nothing inherently bad about email communication with the editor. It may just be that he or she is just old fashioned. These days I still get asked by some editors from reputed journals for reviews in personal emails and send my response via email, too. Nothing bad about this.

My experience (in pure maths) is that this is normal for journals that are run out of a university - which includes some very good ones - and only the for-profit publishers have polished online interfaces.