Everything TheatreTaxi says is spot on.
The most important things here:
- How old are your children
- How tall are your children
- Do they have existing credits and if so what kind.
If your child is aged between 13 and 16, unless they have huge credits, have a very interesting/unique/quirky look AND are under 5ft tall and look young for age then you are not going to find it easy to find an agent as there is no work for this age group.
Existing credits - by credits I mean a contract in a West End show, named role in TV or Feature Film. Things like TVCs, amateur stage shows, modelling, SA (background) roles are not considered credits by agents.
Agencies like Curtis Brown, United, Independent - the ones with the household names on the roster - don't even bother applying unless you landed the child lead in Jurassic Park VI or a major TV production.
For children, the vast majority of childrens agencies will get access to the same castings (a few exceptions like SYA where they cast groups of kids).
So, your decision is whether you want a huge agency or a small one. If an agency can only submit a maximum of five 8-year-old blonde girls who can dance - is your child going to be one of those 5, or do they have 25 to pick from and at least 10 of those are going to be more attractive to a production because they already have experience?
If you want boutique, then in no particular order Allen & Abel, Ardent, Purple Giraffe, Daisy & Dukes, AM Kidz, Young Actors Agency are all very reputable and have some very successful clients.
Of the larger ones, SYA, TTA and Mark Jermin would be good picks.
Or do you actually just want to have fun doing every TV commercial casting where the world and his goldfish is asked to tape, and/or some SA work where you can have fun on a set and enjoy looking out for your child in the background of the next big movie? If so there are agencies that just cater for that.
(10 years experience in the industry, DD has West End and FF credits and has been with agencies listed here, and is now with a high tier adult agent. Many friends in the industry after 10 years so have heard all the good, the bad and the ugly on everything agent/casting/training along the way.)