My aunt and uncle are living in Mexico. They have legal resident status in the US and return periodically so they don't lose it. They would like to start a small business importing foodstuffs into the US from Mexico. What would be the best way to go about this?

Would it be preferable to set up as a private business, a partnership, or as a corporation (and what type of corporation)? Are there certain states where they could incorporate even if they live outside of the US? What would the tax implications be of each method?

Where could I research this further?

I suggest you to seek legal advice
From a Registered Attorney in the State
You are referring to.
Make sure he deals in these matters.