Nope. a chiller isn't needed. just use 2 totes stacked in top of each other and have the spray manifold in the top one and the water in the bottom one. have a submersible pump come up and out the top side of the tote, connected back in with a tube to the bottom of the upper to the manifold. use an ebb and flow 3/4" drain to use for the spraying water that collects in the top to drain back down into the res. it works great and my water stays cool.
Pro tip. ABSOLUTELY 100% Use a filter bag on your pump to avoid clogging your sprayers. this is a necessity.