AudioWorkletで録音します。

RecorderNode.js

export default class RecorderNode extends AudioWorkletNode {
  constructor(audioCtx){
    super(audioCtx, 'recorder-processor');
    this.buffers = []; //processorから受け取ったFloat32Arrayを貯める
    this.port.onmessage = e => this.buffers.push(e.data);
  }

  //Float32Arrayの配列を繋げて新しいFloat32Arrayを返す
  mergeBuffers(buffers){
    const sampleLength = buffers.reduce((acc, buf) => acc + buf.length, 0);

    const array = new Float32Array(sampleLength);
    let sampleIdx = 0;
    for (let i = 0; i < buffers.length; i++) {
      for (let j = 0; j < buffers[i].length; j++) {
        samples[sampleIdx] = buffers[i][j];
        sampleIdx++;
      }
    }

    return samples;
  }

  //this.buffersを繋げたFloat32Arrayを取得
  getData(){
    const samples = this.mergeBuffers(this.buffers);
    this.buffers = [];
    return samples;
  }
}

RecorderProcessor.js

class RecorderProcessor extends AudioWorkletProcessor {
  process(inputs){
    const inputData = inputs[0][0];
    if(inputData instanceof Float32Array)
      this.port.postMessage(inputData.subarray());
    return true;
  }
}

registerProcessor('recorder-processor', RecorderProcessor);

index.html

<html>
<script type="module">
import RecorderNode from 'RecorderNode.js';

window.onload = async ()=>{
  const audioCtx = new AudioContext();
  await audioCtx.audioWorklet.addModule('RecorderProcessor.js');

  //マイク入力を取得
  const stream = await navigator.mediaDevices.getUserMedia({video: false, audio: true});
  const streamNode = audioCtx.createMediaStreamSource(stream);
  const recorderNode = new RecorderNode(audioCtx);

  //マイクを接続した時点で録音が始まる
  streamNode.connect(recorderNode).connect(audioCtx.destination);

  //10秒後にAudioBufferSourceNodeで録音したデータを再生
  setTimeout(()=>{
    streamNode.disconnect();
    recorderNode.disconnect();
    const samples = recorderNode.getData(); //録音したデータ(Float32Array)
    const audioBuffer = audioCtx.createBuffer(1, samples.length, audioCtx.sampleRate);
    audioBuffer.copyToChannel(samples, 0);
    const source = audioCtx.createBufferSource();
    source.buffer = audioBuffer;
    source.start();
  }, 10000);
}
</script>
</html>