Push OverlayEntry Widget above keyboard

Maybe you could try one of below:

• add a padding to the page widget;
• add a padding to the text field;

On the 1st you may add padding wrapping the body:

  @override
  Widget build(BuildContext context) {
    return Padding(
      padding: MediaQuery.of(context).viewInsets,
      child: The page ...,
    );
  }

With the 2nd option you may add bottom padding equals to the keyboard size. At the page (StatefulWidget), you may add the didChangeDependencies:

  @override
  void didChangeDependencies() {
    super.didChangeDependencies();
    setState(() {
      _keyboardSize = MediaQuery.of(context).viewInsets.bottom;
    });
  }

And in your TextField, you set the scrollPadding property:

  scrollPadding: EdgeInsets.only(
    bottom: _keyboardSize,
  ),

you can try this if that's what you really need to do :

Widget itemsWidget() {
                      return Container(
                        width: MediaQuery.of(context).size.width,
                        decoration: BoxDecoration(borderRadius: BorderRadius.circular(10)),
                        padding: EdgeInsets.only(left: 12, top: 4, bottom: 4),
                        child: SingleChildScrollView(
                          child: Column(
                            mainAxisSize: MainAxisSize.min,
                            crossAxisAlignment: CrossAxisAlignment.start,
                            children: [
                              TextField(
                                onChanged: (value) => {},
                                onTap: (){
                                  // here you can control the phone's default keyboard
                                  FocusScope.of(context).unfocus();
                                },
                                decoration: const InputDecoration(labelText: 'Search', suffixIcon: Icon(Icons.search)),
                              ),
                              _foundData.isNotEmpty
                                  ? ListView.separated(
                                padding: EdgeInsets.zero,
                                shrinkWrap: true,
                                itemCount: _foundData.length,
                                itemBuilder: (context, index) {
                                  return Text(_foundData[index]['name']);
                                },
                                separatorBuilder: (context, index) {
                                  return Divider();
                                },
                              )
                                  : const Text(
                                'No results found',
                                style: TextStyle(fontSize: 14),
                              ),
                            ],
                          ),
                        ),
                      );
                    }

I think you should allow it to be scrollable

Try this

you can use some of this 

return Scaffold(
      body: SafeArea(
        child: Center(
          child: Stack(

then on your ItemsWidget

return Container(
      padding: const EdgeInsets.all(16.0),
      child: SingleChildScrollView(
        reverse: true,
        child: Form(

Some thing along these lines, i edited your code

import 'package:flutter/material.dart';

class DropDownWidget2 extends StatefulWidget {
  const DropDownWidget2({Key? key}) : super(key: key);

  @override
  State<DropDownWidget2> createState() => _DropDownWidget2State();
}

class _DropDownWidget2State extends State<DropDownWidget2>
    with TickerProviderStateMixin {
  late AnimationController _animationController;
  late Animation<double> _animation;
  bool isOpen = false;

  final List<Map<String, dynamic>> _allData = [
    {"id": 1, "name": "Andy", "age": 29},
    {"id": 2, "name": "Aragon", "age": 40},
    {"id": 3, "name": "Bob", "age": 5},
    {"id": 4, "name": "Barbara", "age": 35},
    {"id": 5, "name": "Candy", "age": 21},
    {"id": 6, "name": "Colin", "age": 55},
    {"id": 7, "name": "Audra", "age": 30},
    {"id": 8, "name": "Banana", "age": 14},
    {"id": 9, "name": "Caversky", "age": 100},
    {"id": 10, "name": "Becky", "age": 32},
  ];

  // This list holds the data for the list view
  List<Map<String, dynamic>> _foundData = [];

  final layerLink = LayerLink();
  OverlayEntry? entry;

  @override
  void initState() {
    _animationController = AnimationController(
        vsync: this, duration: const Duration(milliseconds: 450));
    _animation = Tween(begin: 0.0, end: -.5).animate(
        CurvedAnimation(parent: _animationController, curve: Curves.easeOut));
    _foundData = _allData;

    super.initState();
  }

  @override
  void dispose() {
    _animationController.dispose();
    super.dispose();
  }

  void _handleOnPressed() {
    setState(() {
      isOpen = !isOpen;
      isOpen ? _animationController.forward() : _animationController.reverse();
      showOverLay();
    });
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: SafeArea(
        child: Center(
          child: Stack(
            children: [
              Container(
                alignment: Alignment.center,
                child: CompositedTransformTarget(
                  link: layerLink,
                  child: Container(
                    padding: const EdgeInsets.symmetric(horizontal: 4),
                    decoration: BoxDecoration(
                        border: Border.all(color: Colors.black),
                        borderRadius: BorderRadius.circular(10)),
                    child: Row(
                      mainAxisSize: MainAxisSize.max,
                      mainAxisAlignment: MainAxisAlignment.spaceBetween,
                      children: [
                        const Text('Select one from below'),
                        IconButton(
                          padding: EdgeInsets.zero,
                          iconSize: 18,
                          splashRadius: 10,
                          splashColor: Colors.greenAccent,
                          icon: RotationTransition(
                            turns: _animation,
                            child: const Icon(Icons.expand_more),
                          ),
                          onPressed: () => _handleOnPressed(),
                        )
                      ],
                    ),
                  ),
                ),
              ),
            ],
          ),
        ),
      ),
    );
  }

  Widget itemsWidget() {
    return Container(
        width: MediaQuery.of(context).size.width,
        decoration: BoxDecoration(borderRadius: BorderRadius.circular(10)),
        padding: const EdgeInsets.only(left: 12, top: 4, bottom: 4),
        child: SingleChildScrollView(
          reverse: true,
          child: Column(
            mainAxisSize: MainAxisSize.min,
            crossAxisAlignment: CrossAxisAlignment.start,
            children: [
              TextField(
                onChanged: (value) => {},
                decoration: const InputDecoration(
                    labelText: 'Search', suffixIcon: Icon(Icons.search)),
              ),
              _foundData.isNotEmpty
                  ? ListView.separated(
                      padding: EdgeInsets.zero,
                      shrinkWrap: true,
                      itemCount: _foundData.length,
                      itemBuilder: (context, index) {
                        return Text(_foundData[index]['name']);
                      },
                      separatorBuilder: (context, index) {
                        return const Divider();
                      },
                    )
                  : const Text(
                      'No results found',
                      style: TextStyle(fontSize: 14),
                    ),
            ],
          ),
        ));
  }

  Widget buildOverlay() {
    return Material(
      shape: RoundedRectangleBorder(
        borderRadius: BorderRadius.circular(10),
        side: const BorderSide(color: Colors.black, width: 1),
      ),
      child: AnimatedCrossFade(
        duration: const Duration(microseconds: 500),
        firstChild: Container(), // When you don't want to show menu..
        secondChild: itemsWidget(),
        crossFadeState:
            isOpen ? CrossFadeState.showSecond : CrossFadeState.showFirst,
      ),
    );
  }

  void showOverLay() {
    final overlay = Overlay.of(context)!;
    final renderBox = context.findRenderObject() as RenderBox;
    final size = renderBox.size;

    entry = OverlayEntry(
        builder: (context) => Positioned(
            width: size.width,
            child: CompositedTransformFollower(
                link: layerLink,
                showWhenUnlinked: false,
                offset: Offset(0, size.height),
                child: buildOverlay())));

    overlay.insert(entry!);
  }
}