I created a form with the "Webform" module. In this form I have one field of e-mail type. My question is: how can I get the value of this field when the user submits the form?
I guess I should use the hook_form_FORM_ID_alter?
Yes, you should use hook_form_FORM_ID_alter() (or hook_form_alter(), if you prefer).
When You have $form argument from these hooks, You should set an array as value of $form['#submit'] with name of your function in which you will catch your needed values.
Example:
$form['#submit'] = array('node_form_function_submit');
[...]
function node_form_function_submit($form, $form_state) {
$form_state['values']; // your submitted form values
}