I personally wouldn't....dogs can drink out of swimming pools and be ok (not sure how much water is ok to swallow of course), but that's a much larger volume of water and the chemicals are pretty diluted. You could certainly add a very small amount, and likely be ok if it was something such as pond or pool cleaner.
But as said, I wouldn't. In such a small pool, doesn't it get dirty pretty fast anyway with dirt and grass being tracked in? I know the times I've gotten a kiddie pool for mine they turned the water brown in a matter of hours racing around the yard sopping wet, then jumping back in the pool. I would just dump/drain/siphon out as needed and refill.