Please listen to Amber: Involve the school counselor immediately. Have you done so already? This is exactly the kind of thing that counselors are there for. He may need some sessions one on one with her for a while to let him talk about his fears (and to let the counselor teach him some coping strategies). The counselor needs to be involved because the K teacher has an entire class to handle and teach and cannot give your son absolute one on one attention when he is stressed.
I would ask tomorrow for a meeting with the counselor, the teacher and the principal or vice principal all present; you all need to form a team to get your son through this.
Be aware that the counselor may recommend that your son also see an outside counselor who deals with young kids. Seek out one with experience in separation anxiety. Your son sounds as if he has extreme anxiety, especially with the vomiting; no one should vomit for hours on end because he's just going to school. No one should sleep so little that he's unable to attend school the next day. His anxiety is very serious and he needs more help than you can provide him; please be open to professional help that involves all of you-- you and your husband will probably need to attend some counseling with him, as well.
Be aware, also, that he cannot continue with so little time in school. When you say you've "made it to four days of class this year" do you mean he gets to school four days out of each five-day week, or he has been to K a TOTAL of four days the entire school year so far? I read it as the latter. Either way, very soon you are going to hear from the school that you need to remove him or have him there a certain minimum number of days, or he will have to repeat K anyway next year.
Homeschooliing will make him relaxed and happy, probably. But do you want to set the precedent that he will just stay home for good? If you do not work on this now, with the school and outside professionals, he is going to be homeschooled from now on, permanently. Is that really good for a child with anxiety? Will he ever learn to be in a larger class setting? Even homeschooling does not mean sitting home with mom forever; homeschoolers have classes, events, workshops, field trips with much larger groups of kids. How would he handle that if he can't handle seeing the same set of kids and same teacher each day? Homeschooling just to remove him from the source of the anxiety may not do him any favors.
I'd go in person to the school first thing tomorrow and say you want a meeting the next day with the "team" (plan to meet without your son present, though!) Tomorrow I'd also ask the counselor to have a contact list of professionals you can turn to for help with his anxiety, ready for you to read on Friday at this team meeting. Please be MUCH more assertive with the school about helping you, beyond letting you sit in the classroom. Are the teacher and others aware that he makes himself physically ill every single day you are not in the classroom?
As others note, you have a younger child; please strongly consider preschool for her! It is only a few half-days a week most places but does a great deal to get kids gradually used to being away from home.