Toddlers do sometimes vomit when they're crying hard because their reflux system hasn't matured yet. They do outgrow it.
I wonder if you're asking if there is some way you can reduce the cause for crying? How do you soothe her? I suggest just briefly being sympathetic while acknowledging how she feels and then distract her to a different activity. The more we sympathize the more upset both the child and the parent gets. If you're worried about her she senses that and gets more upset. She doesn't understand why you're worried/anxious but feels that if mommy is anxious there must be something wrong.
Try handling her upsets in a calm, very matter of fact manner. Don't try to get her to talk about why she's upset. If you know why acknowledge it and treat it in an understanding way while reassuring her she will be OK.
Sometimes, with my granddaughter at that age, just sitting down next to her, not touching her, not talking to her, just quietly sitting with her until she regained control of her tears helped. One possible reason for the out of control tears is that she feels out of control. When you try to console her, you're adding another stimulus to a child who is already over stimulated. Your confident presence helps her feel grounded.
My daughter, when she was young, and now both of my grandchildren, have a special toy that they take with them to help them feel more confident. For my granddaughter who is now 10, it's any 2-4 of her Littlest Pet Shop toys that fit into her pocket. For my 7 yo grandson it's either a small stuffed donkey or a small stuffed bulldog.
She won't grow up with this type of behavior. She is acting this way as a way to cope. She may be unconsciously doing it in part for attention, also. We continue to do what works. If crying gets you to stay with her and she's feeling insecure then she will cry. Not so much for attention. It's a way to get her need met. If you can meet that need in another way then she will be more apt to stop the crying.
What happens on the days that she's not upset when you drop her off. Look back at the whole morning. Is it possible that getting out of the house happened more smoothly? Perhaps you weren't as rushed as you are on other days? Possibly you have breakfast together some days but not on others. Perhaps she wanted to wear something different than what you put out? Perhaps she didn't get enough sleep on those rough mornings. Perhaps some mornings the adults are thinking of something else or are tense because of having a tough job to do at work?
Unfortunately, for sensitive kids, changes in routine are difficult. Also they may be more sensitive to other people's moods.
After reading Amy J.'s post, I wondered if she is having a tantrum, perhaps because she doesn't want to stay or there is too much going on in the room and she's immediately overloaded. Is what you're describing different than her temper tantrums? If she's angry when she starts to cry then this may be tantrums. If they're tantrums the sitting quietly, etc. as I suggested above will also work for those. If they're tantrums it's especially important to not try to interact with her until she's calmed herself down. Any form of interaction intensifies a tantruming child's feelings.