So you can formulate this a number of different ways, but the way I would personally do it is:
ich hätte gern die folgenden Tage frei, da ich [xyz] vor habe. (Ist das in Ordnung?)
This translates to “I would like to have the following days off, since I have [xyz] planned. (Is that alright?)”
It’s a matter of preference. I put the question in parentheses to indicate that it’s optional to ask; I personally like to do it, but I’m also a student and generally ask for permission to do things. Others may disagree, especially if you’re a “real adult,” so to speak. That said, if I NEED to take certain days off - and it’s not up for debate - I would say:
an den folgenden Tagen bin ich wegen [xyz] abwesen, und ich hoffe, dass das in Ordnung ist.
This literally translates to: “On the following days I will be absent because of [xyz], and I hope that that’s alright.” It’s more definitive, but like I said, it boils down to preference and occasion.
Your answer - which others have pointed out as flawed - uses a “1:1” translation and has an unnatural-sounding word choice, which is probably why you feel the way you do. (That intuition is a good sign!)
The word “Daten” is more commonly used to mean “data,” and though it can also refer to “dates,” it would be used in a different context. Someone please correct me here, but you could say something like “Liste der Daten meiner Abwesenheit:” (“List of dates on which I’ll be absent:”) followed by a list of dates, but it’s strangely formal and technical given your example email.
Furthermore, you’re missing a definite article “die” (ie you should have “die folgenden Daten”) instead. Or you could say “folgendes” but that’s neither here nor there in this case.
The word “anfordern” also sounds wrong, but I’m not entirely sure why. I would say that it’s also a strange but understandable choice for this context. If your company has an application process for vacation days, you could say “beantragen” but I assumed that this is not the case, since you are asking someone directly. However, if this is the equivalent of “applying” then you could say:
ich würde gern die folgenden Tage zum Urlaub beantragen.
This means: “I would like to apply for the following vacation days.”
I’d like to add, as a native English speaker, that your first email is not cringy and was also perfectly understandable and grammatically correct as far as I can tell. I might, however, guess that you’re not from my country at the very least (USA) based on wording, but honestly that doesn’t matter. There’s a limit as to how colloquial a professional email should get, anyway.
Disclaimer: I also hope that a native German speaker can quickly verify/critique my answers. My background is as a German language learner of many years and having myself completed an internship there (and asking for vacation days, as well) but my wording may also be flawed. So just because it’s the way I would do it doesn’t mean it’s the way I should do it.