It’s not clear to me where this throwing up is occurring. At home? In the car? At the entrance to the school? It’s not in the classroom, since they aren’t seeing it, right? So let’s look at what happens in the 20-30 minutes before the vomiting.
If he’s “fine” at school and you believe/trust the staff, then it’s not school per se.
And it sounds silly, but have you ruled out dietary issues? Because sometimes wheat/dairy and other things cause problems, and maybe being rushed to get out the door adds to it.
What’s the routine beforehand? Are you learning the rigors of getting out the door on time, and perhaps rushing yourself? Maybe he’s picking up on that even though you think it’s all inside your head. Does he need more time to transition? Some kids just don't get their butts in gear easily, so you need to give a 10 minute warning and a 5 minutes warning (which they don’t understand, but they do understand a kitchen timer or a “watch TV until the commercial and then we turn it off” sort of thing). Does he need to get his shoes on 20 minutes before departure and then play for a bit, rather than “Stop what you’re doing, put on your shoes, where’s you backpack? Let’s GO! We’re late!” (Which we ALL feel, by the way – no shame there! But it’s not good for kids so we have to control it.) So maybe you need a new routine every day, not just on school days. Maybe the backpack and the lunch/snack and the shoes and everything else need to be done ahead. Maybe breakfast needs to be 15 minutes earlier. Maybe your things need to be in the car the night before so you are totally organized and calm.
What’s the routine when you get to school? I’m not one for long, drawn-out goodbyes, but maybe more of a transition routine would be good. That might mean you need to get there 10 minutes earlier, especially if you are going on to work. Maybe he needs a special toy or activity, or a place to sit (the art table, a carpet square in the corner, the Duplo box….).
And the first week is always fun because it’s party time. The second week is, “Wait, this is every day?” The 3rd and 4th weeks are, “Geez, I could be home in my pajamas and doing what I want, instead of sitting at circle time.” So there’s definitely a honeymoon period followed by reality for most kids, whether 2.5 or 3.5.