Male onsen fan here. I'm not sure I'd go so far as to say it's a "custom" to shave in an onsen, but yes, some guys do, and there are onsen/sento that sell/supply razors and after-shave skin lotions for that purpose.
The general atmosphere, or at least as I feel it, seems to be that shaving at an onsen is OK, but not required, and so a lot of guys skip it. Many Japanese men's beards come in thinly enough that they can get away without shaving for a day if they're not headed to the office (like when they're on a weekend onsen trip), and I think in the case of going to an onsen with an attached hotel that they're staying at, a lot of guys shave in their rooms either before or after going to the bath.
All that said, I've shaved my beard (using a non-electric razor) at onsen and never gotten any dirty looks for doing so. I think the OP is right, though, in that using an electric razor might be seen as a faux pas, or at least something most other guests would rather you not do, on account of the noise. Not that onsen have to be completely silent, they have electric hair dryers, after all, but in the case of shaving, where a silent, non-electric option is available, I think most patrons would prefer people go that route.
The lack of electric razor-use could just be for practical reasons too. In some onsen, there's only a basket to put your personal items in, so some people might worry about theft/damage to their electric razor. An electric razor is also bigger and bulkier than a non-electric one, making it one more thing to lug to and from the bath and have to carry around any common-use areas like the onsen's cafe, snack bar, massage chair room, etc.
If you absolutely -had- to use an electric razor, I imagine it'd be considered more acceptable at a sento, where there's still at least a little "We're all here to get clean" mentality, as compared to a more relaxation-focused onsen.
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