In 9.4 you can use coalesce and an aggregate filter expression.
SELECT C.id, C.name,
COALESCE(json_agg(E) FILTER (WHERE E.user_id IS NOT NULL), '[]') AS emails
FROM contacts C
LEFT JOIN emails E ON C.id = E.user_id
GROUP BY C.id, C.name
ORDER BY C.id;
The filter expression prevents the aggregate from processing the rows that are null because the left join condition is not met, so you end up with a database null instead of the json [null]. Once you have a database null, then you can use coalesce as usual.
http://www.postgresql.org/docs/9.4/static/sql-expressions.html#SYNTAX-AGGREGATES