Does this Bible verse have the same unfortunate double meaning in American English as it does in British English?
The photo is taken from the 1984 translation of the (US) NIV. It looks as though the UK edition of the 1984 NIV had the same text; but in the 2011 update, the UK edition has ill in place of sick, while the US edition remains unchanged.
I have visions of people bringing little pots of vomit...
The referent is just elided here. You can read this as meaning "sick [friends]" or "sick [relatives]." The reason it sounds odd to British ears is that the current British colloquial usage of "sick" as a euphemism for "vomit" is overpowering any other interpretation for you.
Since that colloquial usage is relatively unheard of in America, you are correct that the phrase sounds less odd to American ears. Perhaps that did influence the translation choices made by the NIV.
Their sick is a noun phrase without any noun and with an adjective functioning as Head of the phrase. These are referred to by The Cambridge Grammar of the English Language as fused Modifier-Head constructions. We most usually see these with a definite article:
Occasionally, however, they can occur with genitive pronouns in Determiner function. In this particular case, a modern reader might find their sick vaguely unidiomatic. However, we often see fused Modifier-Head constructions with sick in combination with the adjective wounded
Here's a couple of examples from printed books:
If captured, permanent medical personnel and chaplains, although detained, will continue to care for their sick and wounded.
We have never made the slightest difference between our own men and confederate prisoners when their sick and wounded have been in our hands.
I personally also find the Original Poster's example awkward. But that's just my personal opinion.
It is an archaic but correct usage:
Sick (n.)
- "those who are sick," Old English seoce, from sick (adj).
sick (adj.)
- "unwell," Old English seoc "ill, diseased, feeble, weak; corrupt; sad, troubled, deeply affected," from Proto-Germanic *seukaz, of uncertain origin.
(Etymonline)
For any colonial friends, it would sound very natural indeed, because of
Bring me your tired, your poor, your ...
huddled masses, et cetera.
It is the same construction, omitting "persons," and implying group.
Note too that "wounded" is also very commonly used in this way. "Bring me your wounded," "Bring the wounded here..."
{Thanks to action movies that such phrases exist outside a limited circle!}
Chiastic, regarding your question I'm surprised it sounds unusual to you - reads totally normally to me in BrE.
(The fact that "sick" can also mean "vomit", or indeed be an adjective, "I'm sick", and nowadays most commonly means "well-executed skateboard move" - so what? English is incredibly ambiguous at every turn, it's not even worth mentioning when you see a sentence which has a humorous alternate meaning, you know?)
(For example, the immediately preceding sentence happens to have a humorous Monty-Python interpretation ... sending word to the "country": you can picture Eric Idle addressing a large green field and trees.)
