If a new-construction home doesn't appraise for the purchase price, can we back out of the contract?

Usually yes, through your appraisal contingency, but read the builder's contract closely, because it was written to protect the builder. The exact contingency language varies from builder to builder, so confirm what yours actually says before you count on it as your exit. That said, a low appraisal is rarely the problem buyers brace for. In our experience, new construction almost always appraises. The builder is selling many homes in the same community, so the appraiser has plenty of recent comparable sales to support the price, and a short appraisal on a brand-new home is something we have essentially never seen. The pricing is backed by the builder's own closed sales down the street. Keep the contingency in place as your safety net and understand your rights under it, but do not lose sleep over the appraisal being the thing that sinks the deal. If you want to walk through how the appraisal and your contingencies fit your specific purchase, that is exactly what the free Roadmap conversation is for: about 20 minutes where we run your real numbers.