Yes, it is grammatically correct, and I think changing "luckily" to "by luck" is an improvement. Let me answer using complete sentences. 

"By chance or luckily, it didn't rain." This is a grammatically correct sentence.

"By chance or by luck, it didn't rain." This is a grammatically correct sentence. As a matter of style, I like it better because of the use of parallel forms.

"By luck or by chance, it didn't rain." I like this even better, because "by luck or by chance" happens to be a stock phrase or a colocation. At least, I think it is, but I might be wrong. A Google Books search shows 660 hits for "by luck or by chance" and only 71 for "by chance or by luck." However, in a general web search, "by chance or by luck" is more frequent.

Finally, as a matter of logic, there is no real difference between "chance" and "luck,." Some would feel that it is redundant to have both, and that it is better prose just to say "Luckily, it didn't rain," "By chance, it didn't rain," or "By luck, it didn't rain."

There is another old stock phrase: "by hap or by hazard." it's old-fashioned and outdated, don't use it. You may see it in old books. Here, "hap" and "hazard" both mean luck, but they go well together because of the sound of the words. They both begin with the same letter (alliteration) and the same vowel sound (assonance).

Greetings,

They both are synonym to each other.

By Luck- By accident

By chance- 1. By accident

                     2. Not Planned

                     3. Through chance.

Hope it would help!

Regards,

Mantra Deepika

It's grammatically correct to say 'by luck', but whether it is appropriate will depend on the context.  Usually it would be  'by a stroke of luck', 'by good fortune', or 'luckily'.

'Fortunately, I ran into* her in town, and I was able to tell her about it.'   [*met her by chance]

'Luckily, I ran into her in town, and I was able to tell her about it.'

'By a stroke of luck, I ran into her in town, and I was able to tell her about it.'

I agree with Dan.

And no, we usually would not bother to repeat the preposition, although you can if you want to.