Leaf miners are attacking my chard - what can I do?
In general, not much if you want to keep organic. You could try Neem. But that's more for deterring future attacks rather than stopping the ones currently in progress. Once they're inside the leaf munching away, spraying won't reach them in there.
But take heart, it takes alotta leaf miners to kill a plant. Mostly the damage is cosmetic.
If the problem is with a few leaves, you can remove those without harming the plant.
Where I used to live, I had trouble with spring-planted spinach due to leaf miners. So I planted beets in spring as a test crop for my Fall Spinach. When the new leaves on the beet tops stopped showing leaf miner damage, I would plant my spinach. It did much better as a fall crop.