You may, and it comes down to whether the facility met its duty to your child. Daycares and schools are responsible for supervising children appropriately for their age, maintaining safe grounds and equipment, following staffing and safety standards, and addressing known hazards. When a failure in one of these led to your child's injury, the facility may be responsible.
Not every childhood injury is someone's fault — some are truly accidents. The question is whether reasonable supervision, adequate staffing, or proper maintenance would have prevented this one. The incident reports, staffing and ratio records, inspection results, and any footage are what answer it, which is why obtaining them early matters.
If the school is public, an added wrinkle applies: a much shorter government-claim deadline, even though children often have longer for other claims. Because of that, a prompt, free, and confidential conversation is worthwhile — it can identify whether you have a claim and protect any deadline.