React app for movie search, making AJAX call with input value from another component
So the way you would accomplish this in my experience is to move your state up your component tree to a higher level where each component that requires access to that state can have it via props that you pass in.

So for example your SearchForm component could accept all the values and functions it requires via props. It needs a prop to handle the change of the search input as the user types, a prop to handle the submit as the user clicks the submit button and a prop to show the search input value in the input box.

So, for example:

&lt;SearchForm
    searchInput={this.state.searchInput}
    handleChange={(event) =&gt; this.setState({ searchInput: event.target.value })}
    handleSubmit={() =&gt; this.searchForMovie(this.state.searchInput)}
/&gt;


That means your SearchForm could become something more like:

const SearchForm = (props) =&gt; (
  &lt;form onSubmit={props.handleSubmit}&gt;
    &lt;label&gt;
      Name:
      &lt;input 
        className="movieName" 
        type="text" 
        value={props.searchInput} 
        onChange={props.handleChange}
      /&gt;
    &lt;/label&gt;
    &lt;input type="submit" value="Submit" /&gt;
    &lt;h1&gt;{props.searchInput}&lt;/h1&gt;
  &lt;/form&gt;
);


All your state lives in your App component and now you can use that state to pass on as props to any other child components that may need it. 

Some observations/suggestions:


As you can see above there is a searchForMovie() function, I would extract the logic from your componentDidMount hook to a function like this so you can call it whenever needed with the search input you desire
Also consider some error handling on your axios.get() function call as if anything goes wrong with the request you won't know about it because the promise will swallow that (can be confusing when debugging)
The above does not consider any performance optimisations and the amount of renders/rerenders on a large app may be problematic, but my guess is that would be premature for your app and important is first that you get it to work
For your App class I would advise against extending any other class aside from React.Component as that is all you require (if I understand your question correctly)
Also, if you are using this in your class methods you will need to remember to bind them in your constructor to avoid hitting the unknown function error, i.e. this.searchForMovie = this.searchForMovie.bind(this);