You will need permission to record copyright material, you've not said. - it's not quite as easy as sticking a mic near as you need "balance" between the song and the music, normally this means several mics and a mixer to get "balance" - in turn this means long cables, using balanced circuits.
One mic? It may be enough for a "family recording" e.g a school concert, but expect the audience to be noisy - and the hall will echo.
If you buy a camcorder, get a decent tall tripod - and an extra battery + Memory card. A tripod with "Fluid head" is useful for general filming afterwards.
Sony has many models and it depends on yr Budget - if you can get one with "Manual focus" as this stops it shifting during recording - Do check your external mic will work directly - do not use plug-in adaptors - Ideally you need an external recorder with a level-meter and headphones ( Closed-back, large muffs ). But all of this pushes up the Budget - and you'll need to know how to use it.

Practice for a few weeks before the big day.

Hope that helps.