Is there a way to animate the icon of the ElevatedButton.icon() in flutter?

You can use AnimatedBuilder inside elevated button by passing as icon widget. Below I tried code when elevated button clicked the animation starts and will load until button is clicked again. you can control animated icon when to stop by making loader false and setstate it.

You can preview or play with code here

import 'package:flutter/material.dart';
import 'dart:math' as math;
void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      debugShowCheckedModeBanner: false,
      theme: ThemeData(
        colorScheme: ColorScheme.fromSeed(seedColor: Colors.deepPurple),
        useMaterial3: true,
      ),
      home: const MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  final String title;

  const MyHomePage({
    Key? key,
    required this.title,
  }) : super(key: key);

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> with SingleTickerProviderStateMixin{
  
  late AnimationController _countroller;
  bool isLoading = false;
  
  @override
  void initState() {
    super.initState();
    _countroller = AnimationController(duration: const Duration(seconds:1), vsync: this,)..repeat();
  }
  

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        backgroundColor: Theme.of(context).colorScheme.inversePrimary,
        title: Text(widget.title),
      ),
      body: Center(
        child: ElevatedButton.icon(
     onPressed: () {
       setState(() {
    isLoading = !isLoading;
  });
     },
     icon: isLoading ? AnimatedBuilder(
            animation: _countroller,
              builder: (BuildContext context, Widget? child){
                return Transform.rotate(
                  angle: _countroller.value * 2.0 * math.pi,
                  child: const Icon(
                        Icons.sync,
                        size: 40,
                      ),
                );
              }
            )
              : const Icon(
                        Icons.sync,
                        size: 40,
                      ),
     label: const Padding(
     padding: EdgeInsets.symmetric(horizontal: 8.0),
         child: Text("Sincronizza"),
     )
), 
      ),
    );
  }}