So I've been reading these posts talking about new recruits going to sign their contracts and then the MOS they want is full... then they go "well I didn't want to join unless I can get that MOS" and all of a sudden there is an open slot...

Is there a website that people can check on how many MOS positions open up at the beginning of the fiscal year or one that is kept up to date through the year?

no

There's this thing called "Priority MOS" that we get for those that fail out of their language here at DLI that they get to pick from to reclass into. It consists of some of the crappiest jobs I can think of in the Army. Usually when you go to MEPS the recruiters push for those jobs because they have to meet some quotas for them. Usually they can convince most younger recruits to take those jobs, but when they get someone that is determined to get a certain job, they'll usually give it to them after some persistance. They're just doing what they can to fill the slots that the army needs, but in the end, the army just needs soldiers, period.

so when I go to sign my final contract before BCT and they ask me what MOS I would like.. if I tell them 15 Tango and then it is "full" and they try to convince me to sign for something else... would a slot suddenly open if I tell them "well see yah later!"

maybe but not necessarily