Did Jesus use non-believers to perform acts of community service on his behalf? I'm trying to answer this question, "What would Jesus do?"

An organizations I am involved with is a network of foster homes that is used for emergency placement. These are the families that take kids that have been very recently removed from their biological homes.  Foster homes are aggressively recruited from the local church community... but we, the church community, have failed to step up and meet the need.  There are families that are interested in being a part of our program, but they are not Christians.

So the question is.  Do we place kids with the non-Christian families that are stepping up and interested in participating in the program?

Opinions will vary widely and I can understand why people might say yes or no.  What I'm looking for is any biblical examples of where Jesus used a non-believer to do his good work.
Because your questions has a "What would Jesus do?" flavor, I'm going to answer in keeping:

In Matthew 25:31-46 we find there will be those who show up at the judgement and discover they were an outsider even though they did all sorts of work in the name of Jesus thinking they were an insider. Likewise, there will be outsiders who find out they are insiders.

It seems to me that Jesus is suggesting that kindness to those in need is more important than our sheep/goat categories. Believers should read this and get a gut-check. Are we outsiders thinking we are insiders? Are there people we think are outsiders who are really insiders? It sounds possible if you take the parable seriously. I think this can used to establish some priorities in regard to your question.

Additionally, it is possible that some of those outsiders just might come "inside" as they minister "alongside." What greater thing than to make some disciples along the way? It may be best to think of this a secondary aspect of this ministry.

And finally, there is no greater source on this than the Holy Spirit himself. Exercise sanctified discernment when selecting families. The Holy Spirit is able to discern the heart, also regardless of categories. Even Jesus turned some away who seemed like great candidates and accepted others who didn't.