Being the sons of a Thai national, they acquired Thai nationality by birth. What they need now is documentary evidence of their Thai nationality.
If they were born in the USA, the mother should get the Thai birth certificate for them from the Thai embassy or consulate.
Then the boys should travel with the the US passport with the suggested non-O visa to Thailand and subsequently
1. either get free one year extensions at the local immigration office fot the reason of being a Thai national, or
2. get entered in the grand-mother's house registration book, then get the Thai ID card at the district office, then get the the Thai passport at the passport office.
It would probably be too late now to get the Thai passport in the USA for travel on 31 July, it the mother should choose that route for them.