As Vertica supports window functions, something like this can be used to retrieve this data:
select name,
row_number() over (order by name) as rn
from T;
I am not sure how this could be moved into an UPDATE statement though - I don't have a vertica installation available:
update T
set name = name || tx.rn
from (
select id,
row_number() over (order by name) as rn
from T
) as tx
where tx.id = t.id;
I don't know if that qualifies as a "self-join" which isn't allowed. But maybe that points you into the right direction.