Once fully cured, yes. The exposure risk is during application and immediately after, which is why occupants and pets leave during the work and for the manufacturer-stated re-entry period, commonly 24 hours for residential jobs.
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Spray polyurethane foam is mixed on site from two components that react as they are applied. The chemistry of interest is on the A side, which contains isocyanates. During spraying and for a period afterward, the aerosolized product and the off-gassing require respiratory protection and ventilation, which is why installers wear supplied-air respirators and full protection. For occupants the rule is straightforward: leave. The household and any pets vacate during application and stay out for the manufacturer-specified re-entry time. That is commonly 24 hours for residential work and can be longer depending on product and conditions. Ventilation runs during and after. A contractor who does not raise re-entry timing before the job is not running the process correctly. Once cured, the material is an inert plastic. Properly installed and fully cured foam does not continue to off-gas at levels of concern, and it is the same class of material used in refrigerator walls and building panels. The complaints that do occur, a persistent fishy or amine odor weeks after installation, almost always trace to a bad install rather than the product: off-ratio mixing, application in passes thicker than the product allows, spraying onto a substrate outside the temperature range, or spraying onto a damp surface. That is not something ventilation cures, and the remedy is removal and replacement of the affected material. How to reduce the risk to near zero: confirm the installer is mixing on ratio with monitored equipment, ask for the product data sheet and the stated re-entry time in writing, confirm pass thickness against the manufacturer specification, and confirm substrate temperature is in range on the day. All contractors you work with here are trained and insured for this work, and asking these four questions is normal practice. A Climate Zone 5A scheduling note. Substrate temperature is part of a correct install, and the shoulder seasons here produce days where an unconditioned attic or crawl sits below the product range in the morning and above it by mid-afternoon. That is a real local cause of bad installs, and a fair thing to ask a crew how they handle.