검색 input  에  v-model="searchText" 처럼 v-model 양방향 데이터 처리를 해주고

다음과 같이 필터 함수 또는 computed  추가

   const searchText = ref("");
    const filteredTodos = computed(() => {
      if (searchText.value) {
        return todos.value.filter((todo) => {
          return todo.subject.includes(searchText.value);
        });
      }

      return todos.value;
    });

v-for="(todo, index) in filteredTodos"

v-for 목록 처리는  필터 처리된 filteredTodos 를 넣어주면 된다.

<template>
  <div class="container">
    <h2>To-Do List</h2>

    <input
      type="text"
      class="form-control mt-5 mb-5"
      v-model="searchText"
      placeholder="Search"
    />

    <form class="d-flex" @submit.prevent="onSubmit">
      <div class="flex-grow-1 mr-2">
        <input
          type="text"
          class="form-control"
          v-model="todo"
          placeholder="Type new to-do"
        />
      </div>

      <div>
        <button class="btn btn-primary">Add</button>
      </div>
    </form>

    <div v-if="!filteredTodos.length">검색 결과가 없습니다.</div>

    <div
      class="card mt-3"
      v-for="(todo, index) in filteredTodos"
      :key="todo.id"
    >
      <div class="card-body p-2 d-flex align-items-center">
        <div class="form-check flex-grow-1">
          <input
            type="checkbox"
            class="form-check-input"
            v-model="todo.completed"
            @change="toggleTodo(index)"
          />
          <label
            class="form-check-label"
            :class="{ todo: todo.completed ? 'todo' : '' }"
          >
            {{ todo.subject }}
          </label>
        </div>

        <div>
          <button class="btn btn-danger btn-sm">Delete</button>
        </div>
      </div>
    </div>
  </div>
</template>


<script>
import { ref, computed } from "vue";

export default {
  setup() {
    const todos = ref([
      { id: 1, subject: "강아지" },
      { id: 2, subject: "송아지" },
      { id: 3, subject: "고양이" },
    ]);

    const searchText = ref("");
    const filteredTodos = computed(() => {
      if (searchText.value) {
        return todos.value.filter((todo) => {
          return todo.subject.includes(searchText.value);
        });
      }

      return todos.value;
    });

    return {
      todos,
      searchText,
      filteredTodos,
    };
  },
};
</script>

<style scoped>
.blue {
  color: blue;
}
</style>