This can be pretty typical -- but if it's not typical for HER, especially since school is not a brand new thing for her, something could be going on. Likely she will just stop very soon on her own, so I agree with not lunching there and I would add: Make mornings very swift and businesslike; ensure her bag is packed up the previous night, lunch packed and in its bag, every scrap of clothes laid out right down to the socks (and no dithering to try on different clothes). Get her to make the clothing choices, pack the lunch, etc. and treat it like a big, fun privilege. If she dawdles over breakfast, give her X minutes to eat and then the food goes into the fridge and she gets a sliced apple and cheese stick (or whatever) on the way to school.
Basically, try to get her out the door briskly in the mornings, so you need less of the distracting just to keep her going. If you drop her at school (no bus), then do not walk her to her classroom door or even to the door of the school -- stop a certain distance away and make that your "meeting point" for both morning and afternoon. Again, don't let her think that all this is some punishment for her tears--no! Be clear that this is just a way to get to school smoothly and fast and to give her more choices and fun regarding clothes and food. Then drop the topic and just get her there.
With the behavior: So this is new? She wasn't like this with first grade or kindergarten? Does she separate from you willingly and without tears in other situations besides the start of the school day?
Those answers do matter. If this is new and related just to school, just so far this school year, then possibly there's some reason specific to her school day that is making her anxious. If this really is a change compared to her past behavior regarding school, then I would let the teacher know that -- say that you are aware it's pretty typical for some kids to do this but it's not typical for her. The teacher reports that she's fine once you leave and that's great, but I would want to ask about other things in her day, especially if this continues much longer. Because bullying is so much on the radar right now, the teacher might immediately say, "Nobody's bullying her" etc. but this is not necessarily about that; even small things can stress a kid in ways that we adults don't always understand. So think through some possibilities:
--Is the class size a lot larger than last year, so maybe she is feeling lost in the crowd?
--Maybe the teacher (or an aide, or another "specials" teacher in music or gym) scolded a kid in a way that has made her fear that person? (And bear in mind, what a second grader considers to be "He was really yelling at so and so, it was scary" might be no more than a slightly raised voice to an adult, but sometimes kids hear any firmness as "yelling" in their minds.)
--Could she be having problems with other kids? I hesitate to say "bullying" because that gets thrown around a lot for cases that aren't really bullying (targeted and sustained), but it's worth digging into it.
--Is she now separated from some buddies who were in first grade with her and she's not with her friends now, and that has her feeling lost?
--Does she know any older kids, including older siblings, who might have talked up how tough this teacher is, or how hard second grade will be, etc., and she's stressing out anticipating that?
--With the vomiting fear, could she have seen a kid vomit and that freaked her out, or the adult response upset her? She doesn't even have to have seen it herself; some kids by second grade are very adept at telling horror stories to other kids just to freak them out, and if she's sensitive she might have heard about something that has her overly worried now about vomiting or seeing someone else vomit. (Even some adults have a particular horror of vomit, their own or someone else's, so it's not totally unknown to have this kind of fear, but it shouldn't be making her this upset.)
Most importantly, has anything changed outside school to make her clingy toward you in particular? New home/move, marital issues, new sibling, a not-new sibling who needs a lot of attention right now for whatever reason...?
Once you think about whether any of those changes or issues might apply to her, talk to your daughter. Don't do it on a school morning when she's still sleepy and has woken up resistant -- do it at some neutral time when school is a little way off, like on a weekend, and while you and she are sitting at the park; or take her out, just the two of you, for pizza (no siblings around for this conversation). Don't grill her but talk -- Does she like the other kids, does she ever run into Friend who is not in her class this year...
This is all very likely to pass soon, but if you think there is a little something going on, it's worth reassuring her about it. There is really nothing you can do about a large, overwhelming class, or the loss of her buddies who aren't sitting next to her this year, or even a little fear of a new teacher. But you can make clear that you do take her concerns seriously and listen to them--at the same time you get her out the door more swiftly.