Could you please tell me what the diminutive form of reindeer is? How do children call it? A pig is 'piggy', a dog is 'doggy', a reindeer is ... 'reinee'?
The -y suffix (in some words spelled -ey or -ie) is used to create familiar diminutives, usually of people or animals we’re close to, normally domesticated ones. These words may come off as playful or childish, or simply endearing.
It’s subject to various restrictions and these days is not especially productive, at least in formal writing, so you should probable stick to established forms rather than trying to coin your own.
- People: Susie, Johnny, Jamie, Jerry, Billy, Tommy, Willy, Sammy, Penny
- Creatures: foxy, goosey, duckie, doggy, kitty, horsey, piggy, birdy, lambie
One of the restrictions involves syllable counts and stress. Notice how Robert becomes Robbie or Bobby, never *Roberty. Or notice how we say bunny not *rabbity (even though no one now remembers bun). (Also, puppy has its own history: pup is actually a back-formation from it.)
The English-speaking peoples haven’t had the long association with domesticated reindeer that some other peoples have, so they have never been given a short, friendly name such as children might use. There is no word *deery, and *reindeery not only has too many syllables, it might also be taken to mean that something were “like” a reindeer in some fashion.
I can almost imagine a five-year-old calling a little reindeer Rudy thinking of the storied Rudolph the Red-Nosed Reindeer of Christmas fame, but that’s just random creativeness. It’s not an established usage.
See also this question.
As people have mentioned in the comments, there is no stereotypical common noun that young children use to refer to reindeer. I would expect most children either just call it a reindeer, or use some more-or-less idiosyncratic name of their own for it, depending on their stage of linguistic development (I have heard that some children go through a stage where they call all animals "doggies"; see this book for corroboration).
English doesn't really have "diminutives" in the same way that languages like Dutch, Spanish or Polish do. If you were to form a "hypocoristic" or "pet name" based on "reindeer", the most likely strategy would be truncation (because of the syllable count restriction mentioned in tchrist's answer) and suffixation with -y. So, "Reiny" or "Deery".
If you Google "Reiny the Reindeer" you can find some results (e.g. "Noelle Herring on Instagram: 'This is Reiny the reindeer. She lives on our mantel year round because she makes us smile. ...' "; "App Shopper: Reiny The Reindeer - Christmas Holidays Stickers"). But note the context: there are not actually children's terms for a reindeer, but cutesy nicknames for specific personified depictions of reindeer.
I did find one example of "Deery" being used, as a nickname given by an adult for a raindeer toy intended for a young child:
This school project has the main aim to design wooden toys for kids between the ages 4-10. This family of shapes, named the Animinimals (= minimal-shaped animals) has four characters: Piggy- the pig, Beary- the bear, Phanty- the elephant and Deery- the reindeer.
("Animinimals", a project on Behance by Reka Istvan)
Actually, as the author from Hungary, I would guess that she is not a native English speaker, but the nicknames sound natural to me. Note that "Deery" is used here as a proper noun, not as a common noun.