@CandlesInTeacups, is your DS mainly interested in musical theatre, or does he want to audition for TV/film/commercials as well?
If his main focus is musical theatre, most children's MT auditions go out to open call, and it's relatively easy to be seen without an agent. Jessica Ronane, Debbie O'Brien and Verity Naughton are the casting directors most commonly involved in casting MT shows involving children. Jessica has a mailing list you can sign up to, and the others put open calls out on social media.
You usually need to live within the M25 for London-based shows (though there is often more flexibility on this for boys because they are in scarcer supply). Even touring productions and shows based in regional centres such as the Chichester Festival Theatre usually rehearse in London. There are opportunities for children's MT in the north (Leeds, Manchester, etc), but I'm less familiar with the northern circuit as we are London-based.
There are relatively few professional theatre opportunities for straight acting (i.e. not MT) available for children. While the state of screen acting is pretty dire at the moment (ongoing impact of COVID plus the SAG-AFTRA and writers' strokes in the US), I'd say that under-10s are the age group where there is more work available for juvenile actors (though that's not saying a great deal in the current climate!). If your DS is interested in auditioning for professional work more generally, I'd recommend looking at the agencies who belong to the Agents of Young Performers Association. Some agencies cover modelling/stills work as well as stage and screen - I'd avoid those that do unless your DS is genuinely interested in modelling work, as it's a very different side to the industry. As others have said, you should never be charged a fee to join an acting agency. You will need to join Spotlight, the main casting database, as most reputable stage and screen (and some MT) work goes via Spotlight. Children can't join Spotlight unless represented by an agency.
You don't need professional headshots to apply to an agency. A pair of clear head-and-shoulders shots (one smiling and one serious) in good natural lighting and in plain unbranded clothing is absolutely fine. If you are taken on by an agency and need professional headshots, there are plenty of good reasonably-priced options, and I'd be wary of any agency that insists you have to pay ££££ for their preferred photographer.
Worth being aware that there are down sides to having an agent. An agent will submit a child for any work they fit the brief for, regardless if it's work your child (or you) are interested in being involved in. Turning down auditions is the fastest way to annoy an agent - it is in their interests to have clients audition for as many jobs as possible in the hope of securing a job the agent can earn commission from. It means that you as a parent will have very little control over what work your child is submitted for, and very little leeway to turn down auditions without a cast-iron reason (serious illness, bereavement, public exams, age-inappropriate production content). While most first-round auditions are via self-tape these days, commercial work is still often held in-person and at notoriously short notice, so having a job and family set-up that is flexible enough to enable last-minute travel to castings is important.
Good luck to your DS. My own DS was 9 when he started performing professionally - he's an adult now and still acting, and has had some amazing adventures and met some wonderful people along the way.