Without trying to be an echo to what Heidik1725 already said, not sure what you really want to do with these emails, ie, do you mean you want to store them effectively, report on them and provide some kind of stats by "issue type"...or do you mean these doggone things are stacking up like crazy and how are you supposed to handle them without feeling like this is overwhelming?
I have about 20 specific client focus areas right now that all have their own dedicated email resource. This goes up and down, and it all varies in duration, volume, depends on what is going on in that particular "setting". We use some storage/sorting methods similar to HeidiK's, but we are more specific and drilled-down with some than others. Sometimes the issue is very familiar and has an ABC type of response you can pump out, other times, there is some back and forth and you would want to eliminate that symptom by having some prefab phrases and so forth. (Mind you, I am not necessarily endorsing automated stuff that is blended in with one or two lines that reflect the actual experience and 2% of what the emailing customer was looking for--THAT is impersonal, ineffective and shabby, and I say that from a personal customer point of view as well. You either want to recognize a customer or you don't.)
I am sort of guessing that you are coming from the "this is overwhelming" perspective and this really comes down to resourcing. If this needs to be covered on the late night shift, what is the reason someone cannot handle these during the day, and does it need to be handled sooner than the forthcoming night shift? We usually have a service level of by end of next day for email, but we often nab these "day of," often within 4 hours. What about team members that are cross-trained and provide consistent handling taking turns at consistent intervals to cover these, and they all have the same response and storage/sorting/pending/closing procedure?
Don't know what your company provides, but if emails are coming through your main website, it sounds like it is a single service with a variety of inquiries within it, as opposed to multiple services with several distinct requirements. In really rough math terms, if it takes up to 5 minutes to get 1 email off the plate, and that's all a person was there for, it sounds like this is a 1.5 or 2 person job to provide super timely service (with training on how to handle consistently). But since I would imagine there are other things to look after, especially during the day, we have a situation where you would want to mix and match, so that you have 4 or 5 people that are able to handle this over the 24-hour day. My instinct tells me you would want some number during the day/busier shift doing their share of these emails, then some number of your night people that are "best in tune" with the day procedures covering off the remainder. And training could happen during an overlap period for the two sets of shifts we're talking about.
Did any of that take you where you needed to go?