The cruise is to Cozumel and Mahogany Bay out of a port in Florida (sorry I don't know which one). I assume having a green card doesn't actually get them an advantage over residents of Vietnam. I see that neither Mexico nor Honduras permit entry to Vietnamese passport holders without a visa. Is there, by chance, an exception for cruises? If getting the visas is indeed required, is it difficult or just a matter of filling out a couple of forms?
The Permanent Resident Card only lets them back into the United States.
Non US Citizens will be traveling on their home Passport like any other trip, that it's a cruise doesn't matter so much. They will need tourist or similar visas for any Ports of Call.
Example: Required Travel Documentation
Barring any unusual circumstances, getting Visas is usually just a bureaucratic procedure, some forms, some time, and a check. You should visit the website of the Mexican and Honduran consulate for all the details. (Note, the US PRC allows visa free entry to Mexico, but not Honduras.)
You might want to consider an itinerary that includes only countries that allow Visa-free entry to PRC holders. Countries that Offer a Visa Waiver for US Green Card Holders
Also, the major cruise lines partner with Visa agencies that can facilitate the process. The link will be on your line's Travel Documents page.
You will occasionally find exotic itineraries where if specific Visas cannot be arranged, the passenger will just not be allowed off the ship. The country where the ship makes port has to allow this. On a Caribbean milk-run, the cruise line itself might not want to deal with this.