You should say "an L2 learner". Here's the explanation:
Most students think they should only use "an" before the words that begin with the letters "a, e, i, o, u"; however, that is not always the case. The choice between "a" and "an" really depends on the sound with which a word begins and not on the actual letter.
In the case of "L2", you pronounce it as "el 2", so you see, the word begins with the "e" sound (e as in bed). So you use "an".
For a similar reason, you don't use "an" before the word "university", because although the word begins with the letter "u", the word "university" in fact begins with the sound "y", and we don't use "an" before the sound "y".
Hope that helps.