Some legit work at home jobs a stay at home mom could do are blogging for yourself on a topic you know a lot about and want to help people with the content you write, you can also write articles for others peoples blogs or do other freelance work through freelancer.
Other jobs you could do are online jobs such as data entry or medical coding or other paid to write jobs that you can find online or sell stuff through an ebay or shopify store.
Or create your own online store and do drop shipping of products where you take the orders for the products and then pay the drop shipper for their product, the drop shipper then ships the product to the customer and you make a profit.