Since 雪 was already mentioned in the question, the 2nd が can (or must) be changed to は
Yes that would be the first rule you learn regarding the が/は distinction, but unfortunately, there are more rules to consider. See this question for a starter: What's the difference between wa (は) and ga (が)?
In a simple sentence, there can be two は's when contrastive-は is involved, and there can be two が's when exhaustive listing-が is involved:
He can speak English.
彼は英語はできます。 (bold: contrastive-は)
He (at least) can speak English (but he has a certain shortcoming).
彼が英語ができます。 (bold: exhaustive-listing-が)
(Among them,) It is he who can speak English.
In your example, 日本海側は雪がたくさん降ります is the neutral sentence which should be said when there is no prior context. But in this context you are comparing two regions, so exhaustive-listing-が should be used in place of the plain thematic-は.
On the Sea of Japan side, we get a lot of snow.
(Use this if there is no prior context.)
It is on the Sea of Japan side that we get a lot of snow.
(Use this as a reply for your initial question.)
Regarding snow, we get a lot of it on the Sea of Japan side.
(This is also fine as a reply, but now you're changing the topic from 日本海側で to 雪. In other words, if you feel the current topic is "Where does it snow in Japan" rather than "What is the meteorological difference between the two sides of Japan", you can say this.)
Finally, of course there can be lots of が's in compound or complex sentences (e.g., sentences that have relative clauses, sentences with multiple clauses joined with te-form, etc).