Which one is more natural?
I want to introduce Japanese food to her.
I want to introduce her to Japanese food.
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Both versions are grammatical, and at first glance both are idiomatic... but then you start thinking about how you'd go about the introductions in the first version. "Esther, meet sushi; sushi, meet the girl who is gonna eat you." :-)
By all rights, the second version ought to evoke the same reaction. The fact that it doesn't is a clue that it is, indeed, idiomatic, in both senses of the word: it's something people say, and its meaning is more than the sum of its parts.
Note that this restriction only holds for using "introduce" with inanimate objects. If you're using "introduce" in its literal meaning of "Bob, I'd like you to meet Joe", then you can put the subjects in either order, although there's a pretty subtle difference in meaning.
I'd like to introduce my mother to her.
I'd like to introduce her to my mother.
It's kind of hard to articulate the difference, but I think in the first version, Mom is the visitor, while in the second, we're at Mom's house, perhaps not literally, but in intent.
Another wrinkle is that if both introduced items are inanimate, then you have to pay attention to which is the container and which is the added item.
I'd like to introduce Japanese food to her diet.
*I'd like to introduce her diet to Japanese food.
The second version again brings up strange visions of anthropomorphized food items, and is not something most people would say. If you choose different inanimate objects, though, either order can work:
I'd like to introduce Japanese food to her palate.
I'd like to introduce her palate to Japanese food.
I was always taught, in a one-on-one setting, that one introduces a junior to a senior
Please allow me to introduce...
I would like to introduce...
or a person to a thing or group
I wanted to introduce my friend to Modern Art
She introduced me to dancing
Introducing Her Majesty The Queen!
Your second choice would be better.