Ввод выберите и кнопку на той же строке
<div class="form-inline">
<div class="form-group">
<div class="input-group">
<input type="text" class="form-control" v-model="ClienteBusca.value" placeholder="Buscar cliente">
<span class="input-group-btn">
<button type="button" v-on:click.prevent="buscarCliente" class="btn btn-primary">Buscar</button>
</span>
</div>
</div>
<div class="form-group">
<select class="form-control" id="selectedCliente">
<option v-for="cliente in Clientes" v-bind:value="cliente">{{cliente.nome}}</option>
</select>
</div>
</div>
Cruel Capybara