Definitely agree with letters not being needed until secondaries, though you should try to get them in asap in case you have a slow letter writer.
I think you can submit it if all your transcripts aren't in, but it will not be verified. I'm currently having a problem with this because I (unfortunately) went to a school for 1 semester that has a serious problem sending transcripts.
I have paid for 4 transcript requests to AMCAS since May 4th from them, and still they haven't sent it (or claim they have, but AMCAS never received it). I have a hard time believing all 4 were lost in the mail when my other 3 transcripts from other schools made it with time to spare. I'm to the point now that I don't know what to do, as I spent so much time preparing to submit and be verified June 1st and now I can't. I'm starting to worry they will never send it, no matter how much I pay, or how many times I request it.
Anyway, sorry for the rant. In the end, you'll be fine submitting without letters 🙂