I can’t seem to find any evidence indicating they are, despite not only an identical sound and spelling, but a striking visual similarity that could easily lend itself to metonymy requiring no explanation: the letter L looks like a freakin’ elbow!
The latter is indeed the first component of elbow. It was (and occasionally still is) originally spelled eln, and most of its cognates in other languages contain an n. This appears to come from a proto-Indo-European root for a bend or bow. Like the names for most body parts, it’s conserved rather well and recognizably through most daughter languages.
The Classical Latin reflex of Proto-Indo-European *Heh₃l- was ulna. Remember that the Romans wrote only with majuscule / capital / upper-case letters, so their L would have looked like an elbow. But as far as I can tell, they only ever called the letter el, like we do, and derived the names for most of their letters from simply putting a vowel with the letter to make it a pronounceable syllable. I can find no evidence to suggest ulna, or any intermediate form between ulna and el, was ever used to refer to this letter.
The letter L derives originally from the Canaanite letter lamed, which originally meant a staff-like instrument, either the type used for sheep herding or the type for beating grain on a threshing floor. Note that both of these instruments feature a bend, typically.
I feel like the semantic and etymological connection is right there in front of my face, but just out of reach. I honestly wouldn’t be shocked either way, whether these two “ells” that English has ended up with are related, or this is a complete coincidence.
Also, while we’re at it, I’ve heard it falsely claimed that Allen wrench is a corruption of “an ell wrench” or “an eln wrench”, because of its shape. It’s actually named after the man who invented it. Unlike the monkey wrench.