Larger churches may be able to make small donations to community-based projects. If not, see if they'll let you have a bake sale to benefit the playground before or after services.
Churches have many clubs within. Connect with a few to see if they'd like to be involved in your project. Maybe the choir could sing at your ribbon-cutting ceremony, and maybe youth from the youth group are looking for service opportunities.
Church bulletins are very effective at spreading your playspace message! Ask them to run an ad/article about your project, featuring volunteer and product needs.
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