Try a daycare center. Some allow employees to bring their child with them as a benefit.

OP, these days you do not need an agency to find a great job. You can use Care.com or a nanny/mom facebook page. Put togther a great resume and be ready to get a position yourself. So many parents love when a nanny knows herself and the position she is looking for, it shows great strength. Also most employers these days know they do not have to pay the crazy fee agencies ask for.

I’ve only seen one, but the situation was very specific. There was a two-year-old, they had to recently decided not to have kids, and they decided that they wanted to switch nannies and find someone with a potty-trained daughter within 6 months of her age. The thought was that both girls would follow the same rules, it would be long-term and they’d be like sisters. The interview was in two parts, first nanny, then later, girls meeting and playing side-by-side.

An agency won’t help you, OP.

Try care.com.

I really doubt you'd be able to get a job through an agency that allows you to bring your own child along.
Parents working through agencies are paying a higher amount. Most parents who are willing to allow a nanny to bring her own child, are expecting a hefty discount for that privilege.

Also, I'm guessing that having your own child along might cause some legal/liability issues for the agency that they probably don't want to mess with.

Call an agency.

I want to find a job where I could take my kid with me. Can I find a job through Agency?
The current family allow me to have my kid with my at my job. Unfortunately they are moving.
I am devosted to find a new job. Could Agency help me?