Glad it worked out so well.

OP here. I spoke with MB when she arrived home and she said that she swore she had emailed me last week about my vacation time. She approved it and apologized.
She told me never to hesitate emailing her if she's not on the ball about getting back to me.
Thanks for all your suggestions!!

OP here. I have talked to DB about stuff like this and he refers me to MB. She is definitely the go-to person for this stuff. I'm going to speak with her this afternoon.
I plan to ask what steps she would like me to take in the future, should she forget to respond to my email.
Thank you everyone!

^^ If you haven't used DB as a point of authority, it wouldn't be a good time to start now. My husband is a very great dad, but he is not to be trusted regarding the children's schedules, etc... He just gets way too scatter brained. All kinds of hell would break loose if all of sudden the nannies started to go to him. He also knows not to screw around with the schedule unless he wants to be in the dog house. OP, just kindly tell the mom you need to know. She's being very unreasonable and selfish. You've given her notice. There's nothing wrong with reiterating.

If there's a DB in the picture why don't you ask him? Dad's are just as capable of making scheduling decisions.

I'm guessing she needed to check calendars before she could respond, and then forgot (a step she wouldn't need to take with baby emails). If you see her tonight, ask her what she would like you to do in the future. Tell her that you won't (of course) send her lots of emails, and you know her work emails need answers first, but that you're not sure what to do when you don't get a response. Ask her if you should send a reminds after 2 days, or call her, or what she would like you to do.

Why not ask DB about the dates?

Thank you PP. I should have mentioned that she is gone before I arrive and gets home after I leave.
I see her in person about 1-2 times per month.
DB actually just told me that she would be home before him today, so I will see her.
She is incredibly nice and usually very accommodating, so I'm guessing she forgot. I'm going to bring it up to her this afternoon.

If she has time to respond to baby related emails, she has time to respond to your other ones. She is making a choice that they aren't as important. Ask her again, in person, and if she can't give an answer, ask her if she could please find out and let you know tomorrow. Then ask again. At another time I would bring up the slow response issue, and ask her which method of communication that she prefers, and see if you can both agree to an appropriate response time. If she sent you an email, 8 days to respond would likely be unacceptable.

I have been with my NF for 6 months. I am so, so happy with my job!
The only (seriously, the only) thing that I dislike about my job is the fact that it takes MB at minimum a week to respond to any non-baby related email.
I sent an email asking about 2 vacation dates over 2 weeks ago. She confirmed one of the dates and made no mention of the other.
I emailed her back, asking if she could get back to me about the other date, as I have family who's travel plans are dependent upon me being able to have the time off. I told her if it wouldn't work, no big deal. I can figure out a different time. This was 8 days ago and she has yet to answer me. I don't want to hound her but I'm running out of time here. She works a very high-pressure, fast-paced job, so I do realize that it might be hard for her to email, but I feel that 8 days is sufficient time to give me an answer.
I am always flexible, have never said no to coming in early and staying later.
Is there any way that I can tactfully request that she give me an answer?