Add New Footer Widget Area with Limited Options?

Widget areas are the wrong tools for what you need. They are built to offer a choice. Breaking that would be very difficult … and hard to understand for the user.

Alternative: Use a custom setting, for example in wp-admin/options-general.php where the tagline and the site title is. There is even a hook for such additions:

do_settings_sections('general');

First, wait for the action admin_init, and then register a new settings field:

add_filter( 'admin_init', 'wpse_76943_register_setting' );

function wpse_76943_register_setting()
{
    register_setting( 'general', 'footer_text', 'trim' );

    add_settings_field(
        'footer_text',
        'Footer text',
        'wpse_76943_print_input',
        'general',
        'default',
        array ( 'label_for' => 'footer_text' )
    );
}

Easy. Now we need a function to print that field. Use a classic text input field with a class large-text to make it wide enough:

function wpse_76943_print_input()
{
    $value = esc_attr( get_option( 'footer_text' ) );
    print "<input type="text" value="$value" name="footer_text" id='footer_text'
    class="large-text" />
    <span class="description">Text for the footer.</span>";
}

That’s all!

To use the footer text in your theme just check if there is a value, then print it:

if ( $footer_text = get_option( 'footer_text' ) )
    print "<p>$footer_text</p>";

I have written a plugin doing almost the same thing, it is just more flexible: Public Contact Data (Download as ZIP archive).