Is it possible to redirect our customers Shopify cart to a different checkout other than Shopify? So when they click Checkout it goes to a different link and passes the items in their cart to the new url?
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This is an accepted solution.
Yes you could create a custom checkout button that does this, though it would still be possible for someone to go to yourstore.com/checkout directly so you'd need a way to prevent them from checking out that way too.
Shopify has banned checkout hijacks from apps so there isn't really a native offering of this functionality. A checkout button override should work. You could get the cart data from the cart API endpoint, format the data as needed for your new checkout then POST the cart data to that which should automatically redirect them if successful.