Able-bodied, outgoing and accustomed to working, Alexandria Wallace wants to earn a paycheck. But that requires someone to look after her 3-year-old daughter, and Ms. Wallace, a 22-year-old single mother, cannot afford child care.

Last month, she lost her job as a hair stylist after her improvised network of baby sitters frequently failed her, forcing her to miss shifts. She qualifies for a state-run subsidized child care program. But like many other states, Arizona has slashed that program over the last year, relegating Ms. Wallace's daughter, Alaya, to a waiting list of nearly 11,000 eligible children.

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