I submitted a paper to a low to mid-tier online-only journal in my field last January. I did not receive any acknowledgment from the editors that they received my paper and the journal has already published papers that were submitted after I submitted mine. Having been busy since January I forgot to ask the editors for an update. I was only able to send an email a week ago, and I still did not receive a reply. Should I submit my paper to a different journal or should I wait a bit more for a reply?
Give them another week, then decide.
Sending two messages within one week may be considered as pushy, I would try to avoid this. However, if you have serious concerns about the status of the paper it may be justified nevertheless. Whether or not your concerns are serious depends on the history so far: if you have not received any notification at all (via any submission system or similar), then maybe the paper actually got lost.
Personally, I would have contacted the editor earlier since in my field the review duration is 2-3 months -- of course, this may be different in your discipline. Some journals state certain statistics on their web page, such as the time from submission to the first decision. This gives you a hint what time spans you can expect until your paper is reviewed.
Not receiving an email reply after a week isn't unusual, but not receiving any communication for five months is. A short (days) wait for that email might be worth it, but simply waiting generally is unlikely to be fruitful.
I would take steps to prepare to submit elsewhere while you wait for the reply. But if you do decide to submit to a different journal, make sure that you formally withdraw your paper from this one to avoid problems of multiple submissions.
But, it is possible, perhaps even likely, that something has gone wrong and your paper was either submitted incorrectly or somehow lost. You should have gotten at least an acknowledgement that it was submitted.
I would contact the journal first if I were you. Bear in mind, submit one manuscript to two journals at the same time is a big no-no in the academic field.
So contact that journal, ask what happened, or even request to cancel your submission. You haven't heard anything from 5 months, this is highly weird. I've never met this kind of problem.
I agree that 5 months is a long time without any information. Definitely inquire and in the meantime, prepare it for another journal. I would even consider calling someone (editor? administrative personnel?) if you can find a number, but honestly if it comes to that, I would consider submitting it elswhere.