I had a company come in to take my carpets out and put vinyl in. They pulled the carpets out leaving nails exposed and had to leave because the employee brought his six year old that was sick to work and he was puking. I have kids and worried about the exposed nails. The company is acting like it was normal. Are they allowed to leave exposed nails out ?

No, of course they are supposed to finish the job and that means taking all exposed out of the floor. Aside from the nails, you should have been worried about the germs the kid was leaving around. Did the father no know what was wrong with his kid? Get your mad hat on and get going with this company so they know it is not normal. If it was myself, I would be on the phone rather then asking here.