Answer: "Let" is one of those English words that can be translated numerous ways in Spanish, because "let" itself has numerous meanings.
In the example you gave, in most contexts I would probably say, "Quiero apuntar eso," which has a literal meaning of "I want to write that down." If you wish a more precise translation and indeed are seeking permission to take notes, use "Déjame apuntar eso" or "Déjeme apuntar eso," depending on whether you are speaking in the familiar or formal second person, respectively. Dejar is the most common verb meaning "to allow," so what you're saying is "allow me to write that down."
What is important when translating from one language to another is to look for the meaning of what you want to say and translate that rather than attempting to translate words. You simply cannot translate "let" the same way all the time. And if what you mean by "let" is "I want to," then just say the equivalent of that — it's much simpler!
A few of the verbs you can use to translate "let" or phrases that use "let" include liberar (to let go), alquilar (to rent out), avisar (to let someone know), soltar (to let go), fallar (to let down or disappoint), perdonar (to let somebody off, to excuse) and cesar (to let up). It all depends on the meaning of what you're trying to say.
And, of course, in English we use "let" to form first-personal plural commands, as in "let's leave" or "let's sing." In Spanish, that meaning is expressed in a special verb form (the same as the first-person plural subjunctive), as in salgamos and cantemos, respectively.
Finally, Spanish sometimes uses que followed by a verb in the subjunctive to form an indirect command that can be translated using "let," depending on the context. Example: Que vaya él a la oficina. (Have him go to the office, or let him go to the office.)
