You have a number of ways to do this, all with different chances of success.
The Yamaha MX12/4 is a nice, but small mixer, so the snag is simply that it suggests the venue is small, and with only 12 inputs - you have a small snag. There is a pair of connectors that give you a parallel feed of the output - so you can record the stereo mix to a zoom, or any other recorder then sync it with the cameras audio and you have a decent quality result. The snag is that 12 inputs means not everything will be going through the mixer. If the stage is small and the band loud, then even if you have a mic on the bass and guitar, it's quite likely that the PA mixer faders for these will be firmly off, because they are already too loud. In big venues, the bass and guitar get heard by most punters from the PA, not direct from the amps. What people hear in smaller rooms is the sum of everything - the drums, guitar and bass will almost certainly have amplifiers, so the PA just has to get the vocals and anything without amps up to the performance volume. Recording the output of the mixer has these missing so will sound odd. For a folk band where everything is miked up because everything is quiet - the mixer out will be close to the right balance. With a rock band, absolutely NOT! With a sophisticated digital desk - even modest ones, you can hang a laptop onto them with a USB cable and record everything connected to the mixer and sort it out at home/at work. You might need to even pop out a couple of mics for the audience cheering if you want it to sound really good.
there is a different way - most mixers have bus sends available as preface sends. If these are present and unused you could do a mix from these, independent of the PA mix. So you could pop up a drum overhead and put a mic on the bass and guitar and leave those faders off, but use them for the recording. The problem here is simply the mix. If the room is noisy, somebody needs some sealed headphones and needs to do the mix live.
Sounds complicated and to be frank - it is. Recording a mixer out can give really good clarity, but the mix is always compromised in smaller venues.