要在Flutter Web应用程序中使用Google Places Autocomplete API，您需要执行以下步骤：
注册Google Cloud Platform帐户并启用Places API。在Google Cloud Console中创建新项目，并生成API密钥。将google_maps_webservice包添加到pubspec.yaml文件中Flutter项目的依赖项。导入该包并使用GoogleMapsPlaces类向Places API发出请求。以下是如何在Flutter Web应用程序中使用自动完成功能的示例：

import 'package:flutter/material.dart';
import 'package:google_maps_webservice/places.dart';

class PlacesAutocompletePage extends StatefulWidget {
  @override
  _PlacesAutocompletePageState createState() => _PlacesAutocompletePageState();
}

class _PlacesAutocompletePageState extends State<PlacesAutocompletePage> {
  final _placesAutocomplete = GoogleMapsPlaces(apiKey: 'YOUR_API_KEY');
  String _placeId;

  Future<void> _getPlacePredictions() async {
    final predictions = await _placesAutocomplete.autocomplete(
      input: 'Google',
    );

    for (final prediction in predictions.predictions) {
      print(prediction.description);
    }
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text('Places Autocomplete'),
      ),
      body: Center(
        child: RaisedButton(
          onPressed: _getPlacePredictions,
          child: Text('Get Place Predictions'),
        ),
      ),
    );
  }
}