Derived from this question that ended up not being about accordion amplification.
Which options do I have to mic and amplify accordions for live performance? Are dynamic mics on a stand recommendable? How do I accurately capture the instrument?
The only very effective way to amplify an accordion is with multiple microphones attached to the accordion itself. On this picture you can see three, but I've usually seen people using four microphones (two on each side):
Remember that the sound of an accordion does not come from one central spot; each note comes from a different place in a fairly large area. I used to be the accordion player in a very amateurish band, where we used a Shure 57 microphone to amplify it. The very high and low notes were too far from the mic, while the middle range ones were amplified too loudly. With one single microphone, there is no good way to solve that.
Additionally, using one microphone limits you to the right-handed notes only; the bass notes are all off limits, as they come out from the left side of the accordion. Additionally additionally, a microphone on a stand is quite difficult to work with, since the accordion is typically an instrument that doesn't stay in exactly one place.
Sadly, since you're talking about two to four microphones per accordion, this can be a little too pricey for your particular situation. Nonetheless, for the sake of this question's usefulness (which SE is about), I am going to state that this solution is the only really good one.
In case you want to save a lot of money, you can simply hire these things for this one day. I do advise you to test the system before trying to use it on stage though; the more complicated your microphone system gets, the more can go wrong.
Lastly, if you are really not able to spend any money on this, simply use your Shure SM57 microphone. In this case, try to make the accordionist aware of how to play when using a microphone on a stand. Here's a picture of yours truly, where you can see what I mean. (Please do ignore the awesome hat -- let's just say that this is part of some traditional Belgium dress-up culture.)
Firstly, you'll notice that the left hand notes are not being amplified, so there's no use in even playing them. On this picture, I seem to be doing a good job at pretending I am, though. Secondly (and just as importantly), the high notes come out from the lower end of the accordion, and the low notes come out from the top. On the picture you can see that I'm playing high notes, so I'm holding the accordion up a bit higher, aligning the lowest end with the microphone. If the accordionist does not do this, you'll have a very hard time giving every note the same amplification.
Any dynamic mic should work. I have seen accordionists use Sm57s in the past. The instrument is quite loud, and occupies a wide swath of the spectrum. Some folks opt for two microphones - one on each side of the accordian. Some companies sell clip on mics for accordians, but I imagine these are more for convenient movement than for good sound. Some accordian players will put piezo transducers inside the accordian itself. These often go under the bass mechanism, and the grill. Just my opinion, but this probably won't sound as good as a dynamic placed outside the instrument.
You are probably best off with several small condenser mics mounted on the accordion itself. That setup is pretty impervious to movements.
For studio recordings, one might consider big membrane condensor mics in a distance making the movement of the accordion less of an issue.
However, my experience is that this setup works bad for tremolo registrations (I have a quite shallow three-reed tremolo that is quite compelling when listened to directly) which sound much harsher than in reality. My theory is that the large membranes are susceptible to partial vibration modes when excited with the somewhat contradictory nature of tremolo.
It also turns out that tremolo is quite hard to properly reproduce for MP3 at any but the highest quality settings. Ogg/Vorbis degrades quite more gracefully in my experience. But that's a different issue.
The distances for stand-mounted small membrane condensor mics when taking movement into account are unfavorable with regard to noise (recording) and feedback (live).
So there really is not all that much of an alternative to small-membrane instrument-mounted mics. If you are not recording the (moving) bass side, using stand-mounted small membranes might be a consideration. However, it requires discipline.
I have a great solution, playing accordion for years as well as hammond organ often at the same time. I had problems with micing accordions I tried everything to keep up with the rest of the band in terms of volume level before feedback. I used expensive condensers custom mounted to the outside, High end accordion pick-ups but the best thing I ever did and will never go back is to mount a Shure PE56 capusle which is an old sm57 style mic right inside the accordion mounted to the reed block. I have the perfect balance between left and right hand no feedback and even sound throughout did not think this would work at first but my God it is awesome. I play Zydeco music, as well all other types from country to Rock playing keys and accordion and I can tell you I have no longer any problems with the accordion keeping up. I do get a little more key click and noise especially if I smack the bellows shut quickly but hey it is Rock and Roll baby and you can make it part of your sound.
I think the most elegant way to amplify an accordion is by installing an internal microphone system. I work at the Acoustas Co. so I am partial to the system we make however we have tested all available systems internal and external and the AMx11HD and AMx7HD is the most sound solution to amplify an accordion.
I've been using the Thomann ACM-01 pickup system, https://www.thomann.de/gb/thomann_acm01.htm and I am quite happy about it. It has decent sound, it is simple and at a reasonable price. I loved it because I didn't need to do any modification to my instrument - you can always take it off if you don't want it anymore without any damage to the instrument. The difficult part is to figure out an efficient way to install it on your particular instrument making sure the cables don't get in your way. You could combine it with a wireless system, such as https://www.thomann.de/gb/akg_wms_40_mini_dual_instrument.htm, in which case you are totally wireless and amplified.