"request you to give an appointment" or "request you for an appointment"

Question

Which of these sentences are wrong?

This is to request you to give an appointment.

OR

This is to request you for an appointment.

Answer 1

I saw in the comments that the context for this sentence is the opening of a referral letter written by someone requesting an appointment with a physician for their wife.

Both of the sentences in the question sound strange. If I were writing a formal letter of this nature, I would word my opening sentence in one of the following ways:

• I am writing to request an appointment on behalf of my wife <wife's name here>.
• I am <writer's name here> seeking an appointment with you on my wife's behalf.

I included names in my sentences because the opening of a letter is typically an introduction. If they don't apply in this case they can be removed without issue.

Other alternatives if you want to use something closer to the original sentence structure:

• This letter is to request an appointment with you for my wife.
• I am writing you to request an appointment for my wife.

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Why are the original sentences incorrect?

I am not an English teacher, but I can explain why they sound wrong to an American English speaker.

Incorrect: This is to request you to give an appointment.

Correct: This request is to make an appointment with you.

When asking for an appointment with a doctor it is typical to say "I would like to make an appointment." Asking for someone to give an appointment sounds unnatural. Similarly to how you make promises or agreements, you also make appointments. I couldn't find any references explaining the reason, but it could be because you are forming (making) a contract with someone. It is also a two-sided action whereas giving is one-sided and would not have the implication of being newly formed.

Incorrect: This is to request you for an appointment.

Correct: This is to request an appointment with you.

This doesn't sound natural either. The typical context for requesting a specific person for something would be if the establishment has multiple people who could meet with you, but you are requesting someone specific. For example, if you have a rapport with a waitress at a restaurant you may be more likely to request her when you visit. It doesn't make sense in this context where you are specifically reaching out to the person you want to meet with and not the establishment.

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As a side note

If this is truly a letter, I would recommend starting it with some form of greeting and ending with a valediction.

Here are some references to help with letter writing:

• https://www.scribendi.com/academy/articles/how_to_write_a_formal_letter.en.html
• https://www.grammarly.com/blog/how-to-write-a-letter/

Answer 2

Both are idiomatically incorrect.

If you're asking for an appointment, you can say:

This is a request to make an appointment with you

or

This is a request for an appointment with you

If you provide more of the letter or document, we can be more helpful. The way you have it phrased, it's hard to tell if you want to make the appointment with the other person or if you want them to make an appointment with you.