This is an accepted solution.
As stated in our docs, there are multiple different ways of completing a checkout with the Storefront API. The most common way is to use the Web URL / Checkout which loads the Shopify checkout and takes care of payment processing. This will allow you to use any payment gateway that you would usually use with Shopify.
You can read more here: https://help.shopify.com/en/api/storefront-api/guides/checkout-guide
Please have a read through the docs in full. Here is the direct link for Spreedly: https://help.shopify.com/en/api/storefront-api/guides/checkout-guide#spreedly
Take note however that you will require specific permissions for the app in order to utilize this mutation. You can request these permissions from here: https://help.shopify.com/en/api/guides/sales-channel-sdk/getting-started#request-payment-processing (which is linked in the docs too)
Hope that helps!