as an ex-band player of nearly 50 years, i'd say you probably have the makings of the gear you need already.
a small pa would be useful if his mates have the equipment for playing live which would allow him to sing along with the backing of the drums and the amplified bass and guitar so the whole band can play live. posing is as important as singing (Mick Jagger knew that) so he can either sing into the mic hand held or grab the mic stand.
the SM58 is ideal - it was the standard for many years and i still use one live. a balanced lead - 3 pin cannon rather than jack into the pa is best.
for home practice i assume you have BIAB and can create backing tracks. or you can buy karaoke files which BIAB and RB will play.
buy a usb interface like the Behringer U-Phoria UM2 USB Audio Interface which is quite inexpensive. plug the mic and the headphones into the unit and through usb into the pc where you have BIAB and show him how to play BIAB files. he will hear the backing track and his own vocal played back to him. how you set up the nit should be covered by the manual.
you'll hear him sing which will sound weird as only he can hear the backing but he'll be able to practice covers. if the band plays original material get the chords and create a backing track in BIAB
a small pa like the peavey here
https://www.gear4music.com/PA-DJ-and-Lighting/Peavey-Audio-Performer-Pack-Portable-PA-System/C5Jwould be more than enough to play school halls and village halls (which is where we all started) as long as the bass and guitars have their own amps. the drums don't need amplification in small venues. and as the pa comes with mics the bass and guitar players can harmonise behind your sons vocals!
regardless of the schools own set up the band is then independent!