The card says get a stress after the check stress step. I cannot, of course, find the order of things in any document that FFG provides, but my memory isnthat the Activation Phase goes something like this

Reveal dial

Resolve effects from revealed dial (BB-8, AS, etc)

Move

Add or remove stress from movement

Check Pilot Stress

Take an Action if possible

The Lightning Reflexes card says get stress token AFTER the Check Pilot Stress step, but is that before or after I take an Action?

Before, thus the stress will (usually) prevent you taking said action.

from the Rules Reference, p. 4:

1. Reveal Dial
2. Execute Maneuver
   1. Move Ship
   2. Check Pilot Stress
   3. Clean Up
3. Perform Action

"After the Check Pilot Stress step" means immediately after. You'll have the stress before you can act.

It's worth noting that the reference to "Check Pilot Stress" is obsolete now. In the activation sequence laid out in the original core set rules, Check Pilot Stress was not part of Execute Maneuver, but the thing that came after. So you would perform a green, resolve an "after executing a maneuver" ability, then clear a stress. That would have allowed LR to leave you without stress, and they didn't want that, so they made the stress happen after CPS. These days, Check Pilot Stress is part of Execute Maneuver, so Lightning Reflexes doesn't need to reference it to work as intended.