Roughly 35 lbs static. Pretty much nothing for 5/16" Al.
The case that’ll wreck a weekend is if you drive off a platform, the wheel that hits first will see a near-full-weight shock load. For FIRST purposes, you can model that as 5x static, so assuming the full weight of 150lbs is behind it, apply FEA for 750lbs.
I modeled it as an 80mm bearing hold in a 6" plate. I fixed the bolt locations in the friction area around each fastener, and applied 750lbs to the lip on your 5/16" plate in 6061-T6. Maximum stress is ~8x10^7 N/m^2 at artificial islands created by the modeling method, and is still 3x below yield stress (~3x10^8).
Out of curiousity, I re-ran at 3/16" plate thickness, keeping a 1/16" lip for bearing retention, and found the maximum stress in simulation is just about equal to yield stress. Because those maximums are at artificially high stress areas created by the analysis method, I would expect a 3/16" plate to last the season easily.
Then I got really curious, and re-ran it with the 3/8" polycarbonate suggestion (full 1/8" bearing lip). This got some fun displacements… which might actually be to your benefit, the stresses still don’t actually exceed material strength and under “normal operation” you’ll have an easy time keeping all 4 wheels on the ground.
Look into how 2767 designs compliance into their module to get consistent wheel contact & resulting improved module controllability.
IF you are looking for anecdotal evidence, 192 has used 1/4" acrylic and 3/8" wood in their modules without field failures