This same exact issue happened before on another event with Ada in the firing range, not sure if it was last arms week but after interacting with her and collecting the bow on first character alts cannot interact with her to begin the quest. The accept button is there but cannot be clicked on or otherwise activated leaving alt chars unable to progress the event.
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Hello there, Thank you for your report. We understand you're having trouble accepting the Arms Week event quest on your other characters. We have received reports of this and are investigating the underlying cause of this issue. For any updates concerning this issue, we recommend viewing our [url=http://www.x.com/Destiny2Team]X/Twitter page[/url] and our Patch Notes as they become available, posted on the [url=https://www.bungie.net/7/en/News]News page[/url]. Please keep in mind that the Volatile mod (Shattershafts) should still be possible to equip on bows using your other characters once the questline has been completed on the character who initially accepted it.