Generally, how would I go about finding the maximum distributed pressure which a flat rectangular panel of material, say glass or acrylic, can withstand given thickness, length, and width?
For example, a glass panel 10cm by 10cm by 1cm thick. How much pressure in Pa could the panel withstand before failing? Something like a door on a vacuum chamber.
If you wanted to solve this from first principles (difficult - especially for rectangular plates) you would usually use Kirchhoff plate theory, which assumes small deflections ($\delta < t/2$) as well as some other simplifications.
Results have been published for common cases. Roark's formulas for stress and strain has an extensive compilation of plate formulas for different loading and boundary conditions. You can find these published in lots of other places too. For example on Roymech
For your case of a square plate with a uniform load over the entire plate, and assuming simply supported sides:
The maximum stress in the plate is: $$ \sigma_{max} = \frac{0.2874 q}{t^2}$$
where, $t$ is the thickness of the plate, $q$ is the load per unit area, and $ \sigma_{max}$ is the maximum stress in the plate. If we take a very basic failure criteria of $\sigma_{max} = \sigma_{fail}$ then the applied load at failure is:
$$ q_{max} = \frac{\sigma_{fail} t^2}{0.2874 } $$
