How to use Apollo Client with AppSync?

Ok, here is how it worked for me. You'll need to use aws-appsync SDK (https://github.com/awslabs/aws-mobile-appsync-sdk-js) to use Apollo with AppSync. Didn't have to make any other change to make subscription work with AppSync.

Configure ApolloProvider and client:

    // App.js
    import React from 'react';
    import { Platform, StatusBar, StyleSheet, View } from 'react-native';
    import { AppLoading, Asset, Font, Icon } from 'expo';
    import AWSAppSyncClient from 'aws-appsync' // <--------- use this instead of Apollo Client
    import {ApolloProvider} from 'react-apollo' 
    import { Rehydrated } from 'aws-appsync-react' // <--------- Rehydrated is required to work with Apollo

    import config from './aws-exports'
    import { SERVER_ENDPOINT, CHAIN_ID } from 'react-native-dotenv'
    import AppNavigator from './navigation/AppNavigator';

    const client = new AWSAppSyncClient({
      url: config.aws_appsync_graphqlEndpoint,
      region: config.aws_appsync_region,
      auth: {
        type: config.aws_appsync_authenticationType,
        apiKey: config.aws_appsync_apiKey,
        // jwtToken: async () => token, // Required when you use Cognito UserPools OR OpenID Connect. token object is obtained previously
      }
    })


    export default class App extends React.Component {
      render() {
        return <ApolloProvider client={client}>
          <Rehydrated>
            <View style={styles.container}>
              <AppNavigator />
            </View>
          </Rehydrated>  
        </ApolloProvider>
    }

Here is how the subscription in a component looks like:

    <Subscription subscription={gql(onCreateBlog)}>
      {({data, loading})=>{
        return <Text>New Item: {JSON.stringify(data)}</Text>
      }}
    </Subscription>

Just to add a note about the authentication as it took me a while to work this out:

If the authenticationType is "API_KEY" then you have to pass the apiKey as shown in @C.Lee's answer.

  auth: {
    type: config.aws_appsync_authenticationType,
    apiKey: config.aws_appsync_apiKey,
  }

If the authenticationType is "AMAZON_COGNITO_USER_POOLS" then you need the jwkToken, and if you're using Amplify you can do this as

  auth: {
    type: config.aws_appsync_authenticationType,
    jwtToken: async () => {
      const session = await Auth.currentSession();
      return session.getIdToken().getJwtToken();
    }
  }

But if your authenticationType is "AWS_IAM" then you need the following:

  auth: {
    type: AUTH_TYPE.AWS_IAM,
    credentials: () => Auth.currentCredentials()
  }