Format of this meta is OK – it should work. I’m not sure what you’re doing in this query args, though. You use meta query incorrectly.
If you do it like this, it should work fine:
$my_query = new WP_Query(
array(
'author' => $current_user->ID,
'post_type' => 'tribe_events',
'orderby' => 'meta_value',
'order' => 'ASC',
'meta_key' => '_EventStartDate',
'meta_query' => array(
array(
'key' => '_EventStartDate',
'value' => date('Y-m-d') . ' 00:00:00',
'compare' => '>='
)
)
));