Is there a single word for someone who does not eat onions? I remember having heard this word somewhere but do not remember it now.
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An alliaphage is someone who eats garlic (and, by extension, onions), so if you can negate that in some graceful way (analliaphage?), you'll have a word for someone who doesn't eat onions.
Edit: Per the comments below, I suggest the term non-alliaphage for someone who doesn't eat onions.
Background: The terms "akreophagy" (var. "acreophagy") and "aristophagy" were coined in the 19th century to mean anti-flesh-eating.
Allium is the onion genus; the term "alliaphagy" is attested as a term for a garlic- or bulb-eater:
In the same way, you can coin a term for an anti-bulb-eating: "analliaphagy". However this suffers from overgeneralization: it can mean an avoidance of any plant in the allium genus. If you do not mind mixing your Greek and Latin roots (which offends the purists but is still commonplace), then use the Latin word "cepa", "onion": "acepaphagy" or "ancepaphagy".
The corresponding terms for a person who does not eat onions, then, are acepaphage or ancepaphage.
Alliumphobia is the word that you are looking for, but I think alliumphobia describes the phobia in the scent of certain plants including onions.