Can you please tell me what is the food choice / what the food choice is? (inversion)

The inversion has become optional in colloquial English. In the first example you give, there is also a special reason for not inverting: with inversion, a comparatively long and complicated phrase appears in the middle of another constituent. "the food choice for a vegetarian" comes inside "what ... is". Such constructions are difficult to understand and are often avoided in English.

A similar example is "Does that the Moon revolves around the earth at the same rate as it rotates on its axis surprise you?"

Susumu Kuno first studied this phenomenon in English systematically.

You need to find out which is the "subject" and "complement" in a sentence.

In the case of 1."Can you please tell me what is the food choice for a vegetarian?", either what or the food choice can be the subject, i.e.,

What is the food choice for a vegetarian. The food choice for a vegetarian is what. (Both can be regarded as a declarative sentence.)

Note: What is an "interrogative pronoun" and it can function both as a subject and a complement.

In the case of 2 Where the nearest hospital is,

Note: Where is an "interrogative adverb" and it can not function as a subject. That's why the sentence should be inverted in an indirect question.

Hope it helps.