Think it will work?
No.
First of all, your 4013 as you've drawn it won't toggle. But there are lots more problems.
The biggest issue you have is debouncing your pushbutton. The simplest way of doing this is if your switch has a double-throw function. In that case, feed the NO resp. the NC to a Set/Reset FF (your 4013 is fine for this).
After this, you can feed the FF output to a toggle-FF (4013) and then to your MMVs.
Personally, I don't like the approach.
I'd prefer to have the toggle-FF controlling the output, thus being 100% certain that simultaneous triggering of your coils is out of the question.
Each toggle-FF output is gated with a pulse from an MMV clocked simultaneously with the toggle-FF.
Saves you one MMV, but costs you a couple of gates.
But the first thing to think about is switch debouncing.