Customizer options limited to specific user roles?

There are 2 ways to do this.

1. When registering a setting you can use the capability:

$wp_customize->add_setting(
    'display_excerpt_or_full_post',
    array(
        'capability'        => 'edit_theme_options',
        'default'           => 'excerpt',
        'sanitize_callback' => function( $value ) {
            return 'excerpt' === $value || 'full' === $value ? $value : 'excerpt';
        },
    )
);

2. When registering the control, you can use the render_callback argument to determine if the control will be shown or not:

$wp_customize->add_control( 'checkin[theme]', array(
    'label' => __('Theme', 'greet'),
    'section' => 'checkin',
    'type' => 'select',
    'choices'  => array(
            'theme-1' => 'Background',
            'theme-2' => 'Stars (Live)',
            'theme-3' => 'Gradient (Live)',
        ),
    ),
    'render_callback' => function() {
        return current_user_can( 'edit_posts' );
    },
);

Please note that these are completely different.
Using the capability argument in the setting will skip the control altogether and it will not be added to the customizer API.
Using the render_callback argument, the control will still be added in the customizer and it will be available from the API, it will just be hidden.
If hidden (but still exposed in the API) I can open the console and do this to show it – even though you’ve hidden it:

wp.customize.control( 'checkin[theme]' ).activate()