Well, being as I'm kind of in the publishing industry, I'll take a shot. But to chime in on what the others have said, the huge disclaimer is that I'm basically pulling numbers out of thin air. These are nothing more than educated guesses.
First I think we have to define what we mean by "aspiring author". I'm going to define it to eliminate the great sea of idiots who like to tell themselves "one day, I'm going to write a novel" but who, if they're being honest with themselves, know they'll never actually sit down to write a damn word.
I'm also going to eliminate the people who make halfhearted stabs at writing novels, but never finish them.
If we limit ourselves to people who are serious enough about writing to a) complete a manuscript, and b) actually take real steps thereafter to pursue publication, I think we can define "aspiring author." And my guess is, that more or less matches what you have in mind.
So how many of those are there? My gut tells me less than a million in the U.S. The current U.S. population is 322 million people. Have one in 322 people ever actually completed a book? When you account for the children who are too young, et cetera, I have to say I doubt it.
As a cross-check, you can also look at the membership statistics for writers' associations. I'm in Seattle, and around here the PNWA is the main ass'n. I don't have their membership numbers directly available, but am familiar enough with the organization to estimate its membership at about 1000 writers. This is out of a Puget Sound/Washington/Northwest population of right around 10 million people (PNWA draws mainly from the greater Seattle area, but has some folks from neighboring states and whatnot), that's one serious writer per 10,000 people. But of course, not all serious writers in the area join PNWA, so let's just double that and suppose that 1 in 5000 citizens is an aspiring author.
If we presume that the urge to write is not subject to geographic patterns (and why would it be?) then we can estimate the number of aspiring authors nationwide as:
aa = 322000000/5000 = 644000
And really, I'm pretty comfortable with that number. It's probably off by a factor of two or three in either direction, but as ballpark gut guesses go, it feels right enough.
Now we have to estimate the number of publishing contracts that are available. Wikipedia tells us there were 309,957 books published in the U.S. in 2012. Cross-checking against Bowker.com, it looks like that figure is the number of traditionally published titles. Self-published is nearly an order of magnitude larger.
What this means is that roughly 13% of the total output is traditionally published. If we assume that reflects the desires of the aspiring authors (who are the ones that get to choose which publication method to