We would not be able to process submissions from an arbitrary HTML form, but you may be able to collect the same information from the buyer using our standard features, and we can transmit the full order details to your external fulfillment script after checkout.
First, this help page lists the payment services we support for your buyers' checkout payments:
The buyer's name and email address are always collected automatically during checkout, and you can also tick the box to Require buyer's shipping address in the settings of each product to collect that info from the buyer during checkout as well, so there's no need for you to request any of that info by other means.
That just leaves other details you want to collect aside from the buyer's name, email, and street address. For that, you can set up our Variations Describing Item Options and/or Variants affecting item SKU/price/weight/stock feature(s) for the product. Variations can add up to 3 dropdown menus and/or text boxes to the product's Add to Cart button, and Variants can add up to 3 more dropdown menus if needed, and you can even combine those if needed for a total of up to 6 custom fields (only 3 of which can be text boxes) -- see full details on that here:
To have us transmit the full order details to your external fulfillment script by HTTP POST after the buyer's payment is completed, that would use our Webhook Integration method covered here: