Most of the time no. Riding or leading I don't let mine eat.
Banjo in particular, you let him do it only once and it turns into a fully blown bad habit , so I don't let him eat at all when being ridden.
The only time I would was if I actually went for a long ride and jumped off and ate something myself, then I would also let the horse eat.
But I always offer my horses water if the opportunity arrises.