[Vuejs]-How can I make the form editable when I click the edit button?

You should add a data variable isEditable and bind disabled attribute to its value. On click of the button make isEditable = true. This will make your form editable.

<template>
  <div>
    <div class="container">
      <div class="row gutters">
        <div class="col-xl-3 col-lg-3 col-md-12 col-sm-12 col-12">
          <div class="card h-100">
            <div class="card-body">
              <div class="account-settings">
                <div class="user-profile">
                  <div class="user-avatar">
                    <img src="../../assets/people/a2.jpg" alt="Avatar" class="profile_avatar">
                  </div>
                  <h5 class="full_name">abc</h5>
                  <h6 class="email">abc@email.org</h6>
                </div>
              </div>
            </div>
          </div>
        </div>
        <div class="col-xl-9 col-lg-9 col-md-12 col-sm-12 col-12">
          <div class="card h-100">
            <div class="card-body">
              <div class="row gutters">
                <div class="col-xl-12 col-lg-12 col-md-12 col-sm-12 col-12">
                  <h6 class="mb-2 text-primary">Profil Detayı</h6>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <div class="form-group">
                    <label for="first_name">İsim</label>
                    <input type="text" class="form-control" id="first_name" placeholder="İsim" :disabled="!isEditable">
                  </div>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <div class="form-group">
                    <label for="last_name">Soyisim</label>
                    <input type="text" class="form-control" id="last_name" placeholder="Soyisim" :disabled="!isEditable">
                  </div>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <div class="form-group">
                    <label for="username">Username</label>
                    <input type="text" class="form-control" id="username" placeholder="Username" :disabled="!isEditable">
                  </div>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <div class="form-group">
                    <label for="email">Email</label>
                    <input type="email" class="form-control" id="email" placeholder="Email" :disabled="!isEditable">
                  </div>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <div class="form-group">
                    <label for="gsm_number">Telefon</label>
                    <input type="text" class="form-control" id="gsm_number" placeholder="Telefon" :disabled="!isEditable">
                  </div>
                </div>
                <div class="col-xl-6 col-lg-6 col-md-6 col-sm-6 col-12">
                  <label for="gsm_number">Şifre</label>
                  <b-input-group>
                    <b-form-input type="password" class="form-control" id="password" placeholder="Şifre" :disabled="!isEditable"></b-form-input>
                    <template #append>
                      <b-button v-b-modal.modal-prevent-closing variant="default" @click="passwordModal"><i class="fa fa-pencil-square-o"></i> </b-button>
                      <b-modal
                          id="modal-prevent-closing"
                          ref="modal"
                          title="Şifre Değiştir"
                      >
                        <form ref="form" @submit.stop.prevent="handleSubmit">
                          <b-form-group
                              label="Mevcut Şifre"
                              label-for="current_pass"
                              invalid-feedback="Mevcut şifrenizi giriniz"
                          >
                            <b-form-input type="password" class="form-control" id="password"
                            ></b-form-input>
                          </b-form-group>
                          <b-form-group
                              label="Yeni Şifre"
                              label-for="new_pass"
                              invalid-feedback="Yeni şifrenizi giriniz"
                          >
                            <b-form-input type="password" class="form-control" id="new_pass"
                            ></b-form-input>
                          </b-form-group>
                          <b-form-group
                              label="Yeni Şifre Doğrulama"
                              label-for="new_pass_con"
                              invalid-feedback="Yeni şifrenizi tekrar giriniz"
                          >
                            <b-form-input type="password" class="form-control" id="new_pass_con"
                            ></b-form-input>
                          </b-form-group>
                        </form>
                      </b-modal>
                    </template>
                  </b-input-group>
                </div>
              </div>
                  <div class="text-left">
                    <b-button variant="default" @click="editProfile"><i class="fa fa-pencil"></i> </b-button>
                  </div>
            </div>
          </div>
        </div>
      </div>
    </div>
  </div>
</template>

In your component definition

 data(){
  return {
   // Other state
   isEditable: false
  }
 },
 methods: {
  editProfile() {
    this.isEditable = true
  }
 }

if you want the button to behave as a switch, then

editProfile() {
    this.isEditable = !this.isEditable
  }