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If there is a strong need for it to be seamlessly embedded into an external website, then the classic approach would be to pursue embedded forms - in my estimation, this requires a dedicated and experienced web dev team to get up and running, because the setup starts out fairly complex and goes uphill from there.
If you're OK with a little bit of "redirect" trickery, such that it'd be OK for the end user to end up actually on the appian portal page, it would be far easier to set up and deal with (and probably to update), though potentially not as seamless.
Also keep in mind that neither of these methods inherently verify that the user details being passed in actually belong to the user actually passing them. But I will assume you've already taken that into account.
Are there alternative approaches we could consider?
As far as I know those are the only 2 approaches worth even remotely considering. And since Portals came out, I'm unclear how well-supported embedded interfaces might even still be.
Thank you sir