does everyone agree with facewait
Absolutely not.
First, you don't HAVE to send Spring grades before matriculation. You can go through the whole admissions process without your latest grades and give the final Spring grades to the med schools once you're accepted.
Now, if you wanted to include your spring grades after you submitted AMCAS without them, then facewait is right and you will have to update each school individually. It's up to them whether or not they want to recalculate your GPA too.
So, there are more nuances to think about. Will your Spring grades improve your GPA substantially? Will it get you past a psychological hurdle (e.g. 3.28 to 3.31)? Since med schools aren't obligated to update your GPA after they receive your app, then you should make sure Spring grades are on AMCAS in the first place.
If you want to wait for your Spring grades, moreover, med schools don't look at applications until mid-July and so even on the quarter system, your latest grades will be available by then and you'll still be considered with all the applicants who submitted AMCAS on June 3rd.
If your Spring GPA will remain relatively static or if your Spring GPA will hurt you, go ahead and submit your AMCAS without your latest quarter and you'll be fine.
You're under no time pressure, so simply consider Spring's impact on your GPA.