I wish I had a nickel for every time a newbie had asked about using a mixer to record a whole band. The answer is: Yes, you can use a mixer but, before you go buy one, you need to be aware of what exactly it will do for you.
Many standard live mixers do not have the ability to keep all the tracks (instruments) separated for you. Instead they "mix" them all together before going into the recorder, hence the name, mixer. Which is fine if you simply want a quickie on-the-fly mix and don't plan to edit or mix down later. Once those sounds are all combined together, there is no way to separate them again.
If you want the ability to edit, enhance, and remix all those tracks later, you must use "all" separate inputs into your recording machine. This means your interface(s) must have as many inputs as you have instruments and vocals and your recorder must be able to record multiple tracks simultaneously.
Studio mixers, those designed specifically for recording have this ability, but usually not your common live mixers.
