How often do new class seats come out for schools?

Class schedules are published at least two years out.
That's not what is stopping you from reclassing.
MOS classes, like all classes, are divided up by quotas.
95%+ of MOS seats go to MEPS for nonprior service recruits.
You are completing for less than 5% of actual seats.
And those numbers only change when someone with a reservation is cancelled for some reason.

As often as the schoolhouse publishes them.

Seriously, it depends on the school. The more cycles a schoolhouse runs every year (think Air Assault or 11B OSUT, compared to SFAS or 35T AIT) the more often they will publish seats for classes. I went to one AIT that ran a MAXIMUM of 144 per year, and that was if they ran three shifts a day in the sim lab.

It’s pretty sporadic from what my retention NCO explained to me. Class seats can open up at any given time. She’s have to check RETAIN daily for class seats, however HRC reclassification branch periodically sends out news letters which usually informs retention NCOs about class seats opening up.