Yes, you can start applying. The schools requesting reports from LSDAS will recieve updated reports whenever something new is added to your file. Keep in mind, though, that schools may not actually start looking at your file until these letters are in.
I am in the same situation you are, just waiting for my profs to send in the LORs, and I have already started applying. I asked LSAC about this, and the above is the answer they gave me.