Welcome!
Sorry you didn't get any responses earlier; new members often have to wait for a bit for the first five postings to appear.
Your DC sounds like a charming little person, and I'm glad she is busy with the social end of school.
In answer to some of your questions:
1. reading: Yes, reading is the best way to improve vocabulary. Based on what you are reporting now, she will probably fill in the vocabulary gaps pretty quickly over the next year just by reading what she enjoys. I wouldn't really worry about it at all. Was your older child an earlier reader than this one (for example, before K)? If so, that might explain their stronger early performance on vocabulary tests. And, of course, your first child had conventional full-time in-person learning for all of K and 1st, which this child has not.
2. Test info: you have a right as a parent to request any information that is included in your child's permanent or temporary file, including CogAT scores. Typically, a simple written request (such as an email) respectfully requesting a copy of her score report for your records is all it should take. If you don't know where to direct your email, then perhaps a phone call to ask that question would make sense. I would probably start from whichever school administrator you have the best relationship with, and see where they direct you.
