Yes, you could do that, but there are a few things to consider.
First of all, what brand and model is the current mixer that you'll plug into? To get the best sound with least noise, it would be best, if possible, to be able to take a submix of your additional mixer directly after each channel's input trim>EQ>Level. So, the additional mixer should provide a post-EQ>post-channel level output source, possibly to an Auxiliary Out. Actually, if it has an AUX Out (or more) all you really need is post EQ to the Aux, and the output levels should be capable of being controlled by the Aux levels.
What this will do is tap directly after that section to mix together to one (or two) channels to send to the other mixer before the signal is sent to any other circuitry in your additional mixer. This will help keep noise levels (from more circuitry) lower to the input of the main mixer, but still give you control over each channel's volume and EQ. It will submix in the AUX section. (If you want to run stereo, you could send, say, mics/guitars 1-4 thru AUX 1, and mics/guitars 5-8 thru AUX2, if available, and send it to two inputs of the other mixer, panned.) BUT, you need to definitely know the routing capabilities of the new (the additional) mixer. Some are set up differently, as far as pre-post stuff, and some will even let you switch the assignments.
So, the mixer you will add should have at least one output capable of sending post EQ, level-controlled signals PRE-main mix section. (You can still connect amp/speakers and use it for monitoring, if you want. Auxes will just tap out, not stop the flow.) Now then, the mixer you'll go INTO will need some way to get that (those, if stereo) signals past all its preamp circuitry, and into the main mix buss. Why wouldn't you simply plug it (them) into an input channel or two? That's what the unititiated might consider first. A couple of reasons. First, you'd use up an input channel or two, leaving two fewer for the main mixer. Second, you'd be running an already pre-amped and EQ'ed signal thru another preamp and EQ section, adding noise, and possibly over-driving the channel.
So, that's why I asked what the existing mixer is. Does it have at least one (or possibly two) main mix buss returns or insert points, which will bypass all the preamp circuity to inject it directly into the main mix? If it does, that would be ideal. Connected this way just acts like you've added more channel strips to the main mixer, and should (theoretically) give you the best signal with the least noise. I don't know the venue this will be used in, so that's why I asked about possibly two channels and stereo. I don't know what else might be going on at the same time in the main mixer while your 12(?) guitarists are flailing away, either.
The reason I even mentioned it is that I'm imagining a bunch of guitars at one time being mixed to mono. I'm imagining all those same-range notes being mixed down to....indistinguishable mud in both speakers. I'm assuming these are acoustics, or acoustic-electrics? Anyway, the point is that even in a larger space, it might be good to have the ability to place various guitars in different speakers. All guitars have their "voice". Some are bright and zingy, some are round and smooth and even, some are just plain bass heavy. If several guitars are playing at once, they are likely doing different duties. It might help avoid a mud situation if you could direct the zingier guitars out to the left and right, the more even ones between them and the middle, and the guy holding down the rhythm bassy part right in the middle. Actually, if you have a couple of bassy ones, it might not even hurt to have them panned left and right, to keep their more-bassy frequencies from muddying up both speakers when played together. If it's a small room for the recitals, it's kind of like they are playing through a stereo. If it's a gymnasium, etc. and the speakers are by the stage at the end of the basketball court, then the audience will likely be far enough away that if you aim the speakers right, most should be able to hear both speakers adequately. If that becomes a problem, you could always add a couple more speakers and switch one pair's left and right position.
The goal here is not necessarily to display your stereo-separation technique...it's to keep all the guitars that may be going through from walking all over each other, rendering each (and all) indistinguishable. Placing them at separate pan positions will help with that, since this speaker doesn't have to deal with those 2 guitars much, and that speaker doesn't have to deal with these two, etc. Anyway...that's why I asked. Now..we have another consideration.
Tell us about the current situation. The mixer and the rest of the PA. Speaker models, power of the system, mixer model, and how they are currently set up in what room (type, size, material, etc). Are the speakers pole-mounted at the front, etc? Or, are they distributed around the room? basically, give all the info you can. If the speakers are basically just set up in the front, say, on poles... and the mixer is on stage near the performers...you may find advantage in just buying another similar-output powered mixer and another set of similar-type speakers. This also depends on how many guitars you have going on at once. How many DO you have playing at once? You mentioned a bunch, but we don't know if they are just plugged in pre-performance and level-set for convenience so that each can perform solo, duo, etc...or if there will be all of them playing Wild Thing at one time. It may be a combination of those. It might be just as advantageous, soundwise, if all of them are to perform at once to split them up in both pan position AND separate PAs. That way, you may have only a couple guitars fighting for speaker response through any one speaker at once. This may help retain clarity, though it will be tougher to keep track of. Shouldn't be much more difficult, though,if you have a good assignment chart to quickly identify who is going to where..and maybe even have the channels marked by writing on a strip of low-tack tape.
For ANY of this, if you were to try to pan things, you may need someone sitting at the mixer to change pan positions according the performance. If you have a zingy guitarist panned right for a group performance, but he/she is coming up for a solo performance, it would be best to have that guitar panned back to center. If everything was running through two separate PA systems, you could simply use the channel pan control (if the PA is stereo-capable). If the submix routine is used to run one mixer to another, you may have to have someone learn to properly manipulate the Aux 1 and 2 Send level controls to get them panned back to center. It'd be a good mixing lesson for someone interested.
You'd have to have that person presented with something of a "script". "Okay, Rog is up for a solo...pan him center", so the "script" reads "Rog, Mackie 2, CH 3 Pan C" etc., (or adjust Aux 1 and 2 equally, 3/4 way up on level control). Rog finishes. Script reads "Note! Return Rog Mackie 2 CH 3 Pan L 1/4" ...back to 1/4 Left pan (or turn down Aux 2 send level). This will immediately set Rog up for the next number he's participating in, if the performance is planned. If the entire show was planned, and each song has mixer moves, it might be good to try to figure out how to arrange the entire thing for the least possible mixer assignment changes as the show progresses, but the mixer person would have a chart of all the mixer moves to do to be ready for the quickest changes between tunes.
There are a couple other things I could think of, but it won't do any good contemplating anything further until your current situation is laid out more fully, with respect to existing equipment and room, etc., and your intended goal...with respect to the performances (solos, duos, groups with a dozen guitars, etc.) Also, what is your projected budget. That will help with suggestions. Where is your current mixer situated? On stage, or 30-50' away in front?
Anyway, just some considerations. You can definitely run a mixer into another as a sub-mixer. Do you need to? IS there another alternative?
Please feel free to fill in the blanks, and the folks here will conjure up some viable options, keeping all this in mind. We can make it relatively simple and slightly adequate, or a bit more complicated, but better-sounding. It may even relatively simple, better-sounding, AND possibly within budget. We don't yet know. Fill us in, please?