As previous posts here have discussed on several occasions, many grandparents in Phoenix can and often do step in and take over the responsibility for raising their grandchildren. While this may happen because the child's parents have died or are physically unable to care even for themselves, in other cases the issue may be that the parent has, through a series of bad choices, become unable to live the life of a responsible parent.
In the latter case, a parent certainly can work if they so choose, and holding a job and helping their children financially is probably a good thing for the parent. The Arizona Child Support Guidelines even contemplate situations in which children are in the care of grandparents or another relative, or third party.
Basically, when a grandparent has custody, both parents have an obligation to pay their proportionate share of child support. How much they each pay will ordinarily be determined by the guidelines. Basically, the guidelines start by figuring out each parent's income. "Income" has a broad definition for the purposes of child support and can even include items like regular gifts or benefits, such as free room and board. After figuring income, the court will deduct appropriate credits and make other adjustments.
The big difference is that the grandparent can expect both parents to pay their support amount, instead of one parent effectively compensating for their child support obligation by having custody of the child most of the time.
More detailed questions concerning child support, as well as related questions concerning grandparent visitation or custody, are oftentimes best answered by a qualified Arizona family law attorney with experience in grandparents' rights.