Hi, Turtle! I'm a recent VCFA grad, and I experienced plenty of nervousness during the application process. I think it's pretty normal, especially if getting into a program like Vermont's is something you care a lot about. And I think that applying to MFA programs can be a little scarier than applying to other academic programs, since you're being judged almost entirely on something subjective (the quality of your writing) rather than on objective data like standardized test scores and grades. It's reasonable to be nervous! But don't let that hold you back from applying, either--VCFA is a great place, well worth the sheer terror that comes along with putting that application in the mail.
When I applied, I'd been writing seriously for a while but hadn't been published (or even come close). I had only been out of college for a few years, so I asked two writing professors from undergrad for recommendations, but I know other people who successfully submitted recommendations from critique group partners, writing friends, etc. I think it's best to find someone who's familiar with your writing and who can talk about your work honestly and positively, even if that person isn't a teacher or a published author. For my creative writing sample, I submitted 20 pages of a novel excerpt, but I'm sure it's fine to mix and match, especially if you work in more than one genre. Just send whatever work you feel is strongest.
As far as handling the emotional side of things... while I'm obviously biased toward VCFA, there are several other excellent low-residency programs in writing for children out there, and many are structured similarly to the Vermont program. When I was applying, I knew Vermont was my first choice, but I applied to a few other schools as well; that helped me feel a little more confident that at least one of those schools would want to take me on, and I knew I'd be happy at any of them. You might look around a little (or ask people on these boards for recommendations) to see if there are other programs that would meet your needs as well. The application requirements will likely be similar to Vermont's, so you probably won't have to do much extra work.
I know there are other VCFAers on these boards, and I hope they'll jump in to share their own experiences. If you want to talk to current students or alums in more detail, though, you can email the program director, Melissa Fisher, and she'll put you in touch with people who'd be happy to talk. Best of luck, and I hope you go for it!