How would I say the following:
Can I continue studying Japanese?
I was thinking something along the lines of
Watashi wa nihongo o narau no tsudukimasuka?
It's probably wrong, and I could use some help.
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There are three issues here.
tuduku (→ tudukimasu) is intransitive ("to continue [of its own accord]"). You need its transitive sibling tudukeru "to continue [sth.]" (→ tudukemasu).
"Can/May I [verb]?" is usually translated using ～てもいいですか -te mo ii desu ka, here tudukete mo ii desu ka.
nihongo wo narau no is a noun phrase, which should become the object of tudukeru. To make this grammatical you would need another particle wo, giving nihongo wo narau no wo tudukeru. A way to avoid two wos in the same sentence would be to substitute narau no by benkyō, for example. nihongo no benkyō wo tudukeru.
Putting everything together, I would suggest
nihongo no benkyō wo tudukete mo ii desu ka
as an improved version of your sentence.
You haven't given any context, so we can't determine whether this is a natural way of phrasing this question in your context.