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I'd assume you'd want to look into a system like Elavon/Converge that supports EBT/EFT as one of it's payment options.
Usually for any type of payments though, you are not initiating the payment itself from filemaker, but rather using a plugin or API webhook to process a payment through a merchant gateway.
You might want to call 360works to see if their plastic2 plugin supports EBT charges through any of their multiple supported gateways.
I know APIs like Stripe and Braintree are credit card processing only, so you'll most likely need to go to a more advanced gateway like Authorize.net.