I would like to sponsor an immigrant who is a non-family member. How do I get the paperwork to fill this out?
It isn't entirely clear what you mean here by "sponsor" - do you simply mean providing financial support as a joint sponsor on an Affidavit of (financial) Support, or do you actually mean sponsoring someone's immigration case?
In either event, I recommend speaking with an immigration attorney - even for financial-only sponsorship, there are two possible forms, with one carrying very serious legal obligations for the sponsor.
If you are discussing something more than just providing financial support, this is even more complicated - if there is no family relationship, the only way to sponsor someone for a nonimmigrant (temporary) visa or for permanent residence would be through employment - a bona fide (legitimate) offer of a job. Even this will not work in all circumstances, and the specifics of the situation - your ability to offer employment and the person's work and educational background - need to be discussed with an attorney.
An attorney should also review what other options the person might have, independent of your sponsorship.
"Sponsorship" has many meanings in immigration law. So the answer to your question is a definite "IT DEPENDS!"
If an individual owns a business, then it may be possible for the business to "sponsor" a foreign national for a temporary or permanent (immigrant) visa. The rules are complex and the process can take a long time, but it may be possible.
If the foreign national is applying for a visa, and needs documentation of financial support (to show that the person will come to the US and be 'supported' while in the US), then it is potentially possible for you to do this for a non-family member.
If you simply know someone from another country and want to get them a green card based on your personal friendship, no, that is not possible.
You or your friend probably need to consult with a US immigration lawyer to determine what options, if any, exist in this arrangement.