Ok, I see. Apart from custom GraphQL stuff etc., there are two solutions with Pipefy only.
The first solution is pretty much my second proposal: an additional pipe with a portal.
In your example, the purchase team moves the card to phase “awaiting approval”. An E-Mail template is sent to the supplier, which contains the portal link to the second, “approval”, pipe with a prefilled connected “purchase” card (i.e. connected to pipe “purchase”), which is hidden using conditionals. Use an automation: on creation of a card in the “approval” pipe, update the connected “purchase” to the value specified in “approval”.
The second, better solution; if you have access to Pipefy Beta, you can use Public Phase Portals.