You do need to see a doctor about your sickness/vomiting. That can be a sign of excess stomach acid, a bad gall bladder, a stomach ulcer or several other things that all need treatment. Of course that can also be a sign of cancer, which is why it's important that you get checked out (I don't want to scare you, since most forms of cancer are not common in a young person, but since most are very treatable early on, it's important you get checked out).
Are you in college? If you are, you should be able to go to your campus health clinic for free. If they can't help you, let them know that you have no insurance and you need to see someone on the cheap; I'm sure they'll help you find someone.
If you're not in college, do you qualify for state health insurance? State requirements vary a lot, but most are dependant on your income, whether or not your employer offers to cover you, and whether or not you've been turned down for regular insurance. State insurance has to cover you if no one else will take you. And most will take you if your employer offers insurance, but it's too expensive for you to afford.
Call your public health clinic and see what services they offer. I don't know if they all offer a doctor's visit, but I know I went to one to get a vaccine and it only cost me a few dollars because I was working a minimum wage job at the time. They base your pay on your income and it's very reasonable. That same shot was not covered by my insurance and would have cost $75 or so from my doctor. If they have a doctor available, you can try and see him/her. May not be the best doctor in the world, but at least you'll be able to afford any blood work he orders.
If none of those options pan out, then try a regular doctor. It helps if you have one you've seen in the past--they're more sympathetic if they know you. Call his office and tell them how you're sick but that you don't have insurance, so how much is it going to cost? Ask if they'll extend to you the same discount they give a major insurance company if you pay cash. A lot of doctors like getting cash now because they don't have to submit paperwork and wait for weeks to get reimbursed and so they'll go ahead and give you the insurance company rate. There's nothing wrong with shopping around, so don't be afraid to ask for the cost upfront and pick a doctor based on what deal they'll give you.
And should you need medicine, make sure 1) that your doctor knows you are uninsured and that you want an older, generic drug if at all possible 2) gives you any samples he might have and 3) you call around to pharmacies and get their prices. I have heard that there's a HUGE difference between some of them. I noticed last night that my Welchol at Walgreens was $104 but it was only like $88 at the local hometown pharmacy when I went to get it a couple of years ago. It's on my insurance now, so I go to Walgreens because it's on my way home, but if you're paying out of pocket, you have to watch for that. I have heard on serveral occasions that prescription meds the local places are often cheaper (more expensive on OTC stuff, though).