I'm wondering if your little fellow is so anxious and fearful that his stomach is upset and unsettled - thus he vomits. This is a tough one - you need to find out if this child is "afraid" of the mother or the upcoming day with her or perhaps someone else in her life. But, you will need to be so tactful so as not to create issues/suspicions if there are none. What would happen if the father happened to pick him up a couple of hours earlier (unexpectedly) or return that same day with an item that was (supposedly) forgotten?
If you read the related discussion "Throwing up and nerves", Dr. Kennedy describes this issue in relation to separation anxiety. So, whether you are dealing with anxiety (caused from separation) or fearfulness (caused from unsettling times/actions from the mother and/or home situation), I don't know. But, the father really does need to find the answer. All the best ...
Could be his misses his mom and needs to see more of her, what is his behavior like when he sees her, is he happy,who looks after him the rest of the week if you only have him once ?