It depends. If you were looking for coverage for preexisiting condition you would have X days after your initial deposit - if the plan even covers it. I've seen 7,14 and 21 days after in some policies. If you are past these days you may still get insurance however I'm not sure how late. I think I've read prior to final payment however I really do not know.
Editing - I forgot to answer the other question - the who. We are using Travelex however there are many names: Access America, Travel Insured, CSA, Travel guard. There are some really great threads regarding insurance. There was a link to an write up on insurance explaining the different sections which I found very helpful. Try searching using
travel insurance. If I find the link I'll come back and attach it here.