The statement of purpose or personal statement is something that you could start on before SOPHAS opens. Unfortunately you can't fill anything out until September because the website closes over the summer (the actual application part is not functional until it re-opens) and it clears information entered from the year before.
There are specific letters generated by SOPHAS for transcripts and when I applied you entered your LOR writers' information into the system and it generated an automatic email sent to them with a link where they could complete the letter online, that was in 2010, but I'd think it's still the same... you wouldn't be able to get those until the day SOPHAS opens. If SOPHAS hasn't closed yet this year you could make an account just to download the transcript letter to have it early, but it's a lot of work just to get your transcripts in a few days early (they won't accept them until SOPHAS opens in the fall, so you'd have to have good timing, haha). Basically I don't think it's worth it 🙂
The statement of purpose has a limit of 1500 words (85% sure I remember the number right) on SOPHAS, but most schools like for them to be shorter. I spent a day emailing schools well in advance and asking them what length they were looking for in an SOP. All said whatever the SOPHAS cutoff was their cutoff, but they valued conciseness and expressing interests and abilities in a succinct manner, I think all of them had a thesaurus out and used variations of the word "concise" as the main theme 🙂. Point being, starting on your SOP early, getting that really strong, and researching a few very specific things you can put into the SOP for each school would be a good use of your time. I had about 3/4 of my SOP that was general for all schools, and then the last 1/4 was what I specifically liked about that school and why it would help me reach my goals (as well as how I was a good fit for them). Since most schools don't interview, your SOP is one of the most important aspects of your application!
As another poster said, most of the adcoms aren't going to meet until they get a worthwhile number of applications in OR until a set date. When I applied I sent my application in on the earliest possible date (schools got it in early October). Tulane is on rolling admissions and I heard from them within just a couple of weeks (some other schools are on rolling admissions as well, you can usually find out on their websites). I heard from Emory in December.
Another thing to consider: I go to Emory and have learned some about the admissions process. From what I understand, applications are reviewed by individuals on the committee and then the adcom meets just a few times to go over what was found. Their first meeting is right before winter break and then in January and again later. They have a set amount of spots to fill so they only invite a small number of people at that first adcom meeting to make sure they have plenty of spots to admit the most qualified students regardless of when they apply.
Applying early doesn't hurt you at all, but as long as your application is sent out in Oct/early Nov you would be in that first review pile for a lot of schools. So a lot depends on the schools that you are applying to, if they are on rolling admissions then the time of your application getting in matters more than a school like Emory where adcom isn't going to make a decision until later in the cycle.
As long as you start working and sending things in when SOPHAS opens, you are going to be early with your application submission.
Good luck!