The correct word in this context is "its." The sentence should read: "The dog went to the veterinarian for its yearly checkup."
Here's a brief breakdown of the terms you're asking about:
- "its" is a possessive pronoun, indicating that something belongs to it (in this case, the checkup belongs to the dog).
- "it's" is a contraction for "it is" or "it has."
- "they're" is a contraction for "they are."
- "their" is a possessive pronoun used to indicate that something belongs to them.
So, the complete and corrected sentence is: "The dog went to the veterinarian for its yearly checkup."