How do these young actors get auditions for major roles?

(1) Usually they have acted and performed since a very young age.

(2) They attend an acting school that has connections with agencies or respected acting teacher who referred them to an agency.

(3) Their parents understand the business end of the industry and had the industry connections and knowledge to effectively market their child's skill to get an agent.

The point is - companies are not looking to pull the "right" kid off the street and put them in a movie. That's why they go through agent. It's an agent's job to find the kids who have demonstrated they have the talent and the discipline to work at a professional level. This is show BUSINESS. People are in it to make money, not teach some kid how to act in front of a camera. There's more to professional acting then really, really wanting to be in a movie and willing to work hard. Time is money so they need kids who know what is expected of them and can do that.

So a company hires a casting director (CD) to find qualified actors to audition. The CD contacts agents about the types of roles they need. The agent submits actors for the appropriate roles. (To submit means to send in an actor's professional headshot and resume listing their acting training, experience, special skills and links to their show reel to the CD. An agent can submit more then one of their child actors for a role.). If the CD is interested, they invite the actor actor to audition. So even with an agent, it's not easy for child actors to get roles.

For example there are about 20,000 child actors with agents just in LA. For each role you can figure there are 1500-2000 child actors of the same sex and age range submited for that one role. Of those maybe 20 kids will be invited to audition for a TV role. That's it. Since there's more time to cast a movie they might invite maybe 50-100 kids to audition for each role. Still not a lot compared to the number of submissions. Of those only a few who audition will get a "callback". Only one will get the job.

One of two ways - they've been training from a very young age at a good stage school with an agency attached, or they are in some way connected within the industry already. Daniel Radcliffe, for instance, started training at 5 and was working by the age of 10 - and his parents are a literary agent and a casting director. Hardly surprising he stood a good chance of success!

They have been acting as such a young age and usually most of them have in some ways connection to the directing cast. Example would be the roberts. Emma roberts came from an acting background of family members whos father and aunt are popular actors or actresses. So she indeed has a connection to people who knows who to hire

But they mostly HAVE had training - and loads of experience.

Many kids start at excellent stage schools at the age of about 6, perform with youth theatres' junior sections regularly, go to top private schools with superb drama departments.

Other kids have parents who have been in the business for years, so the child has grown up with actors, theatrical types, attending rehearsals and film sets, effectively being taught by professionals from birth.

As tony says, lots of young actors have connections or family members in the business as actors, producers, directors, casting directors, etc, etc.

Or they have been acting for years usually starting as baby/child models then getting into commercials

You mention "Modern Family" -- here are the backgrounds of the kids on the show:

Rico Rodriguez had been acting professionally on screen for 3 years before Modern Family and he had connections as his sister had already been acting before he started. His sister's agent started representing him because she met him once when his mom brought him to a meeting with the agent and couldn't find a babysitter, and the agent thought he was a funny kid and decided to represent him as well

Nolan Gould started as a child model, doing commercials from age 3 and worked non stop ever since

Both of Sarah Hyland's parents are actors, and like Nolan, she got started as a child model at age 4 and started on screen in commercials as a kid

Ariel Winter also started in commercials, with her first national commercial for Cool Whip when she was 4. Her older sister Shanelle Workman is an actress who appeared in the original Broadway production of "Les Miserables" then went on to soap operas. So she had family connections to help her get her first agent

Aubrey Anderson-Emmons has a mother who is an actress and comedian with a monthly show at The Hollywood Improv and a father who is an entertainment industry executive

They have older family members who already have a job in the business.