Scared To Ask For A Letter Of Recommendation

First of all,  it's pretty common to doubt your abilities and only remember the times you messed up. I think more than likely, your preceptor would be happy to write you a letter of recommendation. I agree with the two other responses. Talk to your preceptor about your overall performance face to face before emailing a request for a letter of recommendation.
Hopefully you've been communicating with your preceptor about your interest in working in the NICU throughout your time with them. If not, I'd suggest you do next time you see them. I think it's helpful if a recommender "sees it coming." (They might put more of an effort into noticing specific things or it might give them time to reflect on if they think you are a good fit for the NICU.) When you ask for their honest evaluation, if you aren't getting concrete answers, consider directly asking what they think you do well and one thing you can improve on. I think in most cases, you should be able to tell based on these responses. If the conversation goes well, I'd ask write there and then if they would consider writing you a letter of recommendation in the future. Assuming they say yes, I'd still send that thank you email and let them know how much the experience has impacted you and THEN send a request for a letter of recommendation. I like keeping the gratitude for the experience separate from the ask. That's just me though. Everyone handles this sort of thing differently. Good luck! Let us know how it goes.