They sound like "leaf miners" which seem to love spinach and chard. Pick off the worst leaves and spray with Rotonone. That should rid you of them. At the first sign of a repeat invasion, repeat spraying. Enjoy the chard!

Ron
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Instead of spraying, I'd suggest daily rounds of the plants, then either squish new damage or immediately remove and use/eat the leaves, trimming out the extra protein.

If you do spray, follow the directions exactly as to how strong to mix it and how long to wait between treatment and harvest.

(Because I'm not familiar with uses for rotenone, I looked it up via google. The label lists various veggies but none where you eat the leaves.)

For next year, you can avoid spraying if you plant in a different spot, then immediately loosely blouse row cover over the seedlings/transplants, securing the edges with soil or boards.

The reason to avoid planting in this same bed is because part of the life cycle is in the soil.

Swiss Chard seems to be most attractive to aphids and leaf miners and control of aphids is most easily accomplished with sharp streams of water to knock them off. Since leaf miners are inside the leaf there is nothing you can spray on the plant that will get into that leaf to kill them, so the best control for leaf miners is to cover the plants with floating row covers to keep the fly that lays the eggs that become the larva that mine the leaf from reaching the plants.
Rotenone is not an acceptable organic pesticide today because of its hazardous nature. Although once acceptable new information about this says it should not be used.

The following is information taken from a Rotonone instructional publication -
"Rotonone acts as a stomach poison and as a contact insecticide. Not toxic to honeybees, but will kill some beneficial insects. Registered for use against most chewing insects on many vegetables and some fruits. Different brands and formulas are available for various pests. Both liquid and dust are available commercially. It has been fatal to mammals if inhaled over extended periods. Rotenone is effective against a wide range of insects and has a short residual life."
Used properly, it is organic, highly effective and safe. It breaks down quickly and plants are safe to eat in a day or two. Hand picking infected leaves is a big job, especially if you have a large planting, and highly ineffective because once you see leaf damage from "leaf miners' they are already wide spread. Row covers are effective if used early enough, but if you have to plant under row covers to get a crop, you might just as well plant in a greenhouse. Part of the enjoyment I get from gardening is watching my plants grow and prosper. Somehow, row covers taks a lot of that away for me.

Rotonone acts as a stomach poison and as a contact insecticide. Not toxic to honeybees, but will kill some beneficial insects. Registered for use against most chewing insects on many vegetables and some fruits. Different brands and formulas are available for various pests. Both liquid and dust are available commercially. It has been fatal to mammals if inhaled over extended periods. Rotenone is effective against a wide range of insects and has a short residual life.