At what gap width between two plates does convection not occur? Does the Grashof number lead to the answer? 
The Wikipedia article (https://en.wikipedia.org/wiki/Grashof_number) yields an equation for vertical plates $$Gr_L = \frac{g\beta(T_s-T_\infty L^3)}{\nu^2}$$ Could I just solve for $L$, with $Gr_L$ equal to $10^8$ (upper boundary for laminar flow)?
 A: You're on the right track here.  For convection between plates to 'stop' (it won't really stop, just get wicked slow) you want viscosity to be very high relative to buoyancy, so you want low Grashof.  Also, in the case of very narrow plates, you'd want to use the gap width as your characteristic length $L$.
$10^8$ is the transition point at which the flow goes from being laminar to turbulent.  That's already a fair amount  motion. Unit-wise, the velocity should scale as 
$$\frac{g \beta \Delta T L^2}{\nu}$$.  That'll get you a ballpark idea of the velocities involved.
Also look the Rayleigh number , which will tell you how strong convection is relative to conduction.  As that gets towards, one you'll be seeing really negligible flow. (note : it'll be practically the same as Grashof for gases).
And last an empirical correlation using $w$ as gap width and $L$ as height:
$$Nu_w = \frac{hw}{k} =\left(\frac{576}{[Ra_w(w/L)]^2} + \frac{2.87}{[Ra_w(w/L)]^{1/2}} \right)^{-1/2}$$
Good for two plates at the same constant temp (other cases can be found scattered around the web).
Just get $h$ low enough that can say that convection has stopped.  Again, you won't stop it, you'll just make it really small.
