Learn Hardcore Turkish: Lütfen peyniri rendele.

Because -i is the accusative case marking a definite direct object. Peyniri means the cheese (a specific, known cheese). If you say peynir, it’s indefinite: cheese / some cheese.

Definite: Peyniri rendele. = Grate the cheese.
Indefinite: Peynir rendele. or Biraz peynir rendele. = Grate some cheese.

The vowel in the accusative suffix follows vowel harmony:

Last vowel a/ı → -ı (e.g., yağı “the oil”)
e/i → -i (e.g., peyniri “the cheese”)
o/u → -u (e.g., tuzu “the salt”)
ö/ü → -ü (e.g., sütü “the milk”)

What verb form is rendele?

It’s the second-person singular imperative of rendelemek (to grate). Imperatives tell someone to do something:

2nd sg (informal): rendele
2nd pl / polite singular: rendeleyin
Very formal/polite: rendeleyiniz

Is Lütfen peyniri rendele polite enough?

It’s softened by lütfen, but it’s still a direct command to someone you address as sen (informal). To be more polite/formal:

Polite request: Lütfen peyniri rendeleyebilir misiniz?
Neutral polite (no “please”): Peyniri rendeleyebilir misiniz?
Polite imperative: Lütfen peyniri rendeleyin.