Destiny 2 Solstice of Heroes bounties aren't awarding Bright Dust due to a bug.
Players quickly noticed that their bounties were empty after the event began, and when a Reddit thread on the issue caught fire, the Bungie Help Twitter (and community manager Cozmo) confirmed that the lack of Bright Dust was caused by a bug. The studio is currently looking into a fix, but until this bug is squashed, players should hold onto all of their weekly, daily, and repeatable Solstice bounties.
You can get Bright Dust from all sorts of bounties, but the Solstice bounties are a particularly good and relevant source since everyone is going to be grinding their Solstice of Heroes armor sets for some time. Solstice bounties can be completed in the process, and the Bright Dust they (should) dish out is especially valuable since it can be used to purchase the ornament kits for the fully upgraded Solstice armor.
Each ornament kit costs 6,000 Bright Dust, so if you don't have some stashed away, you'll need to grind out some Bright Dust before the Solstice of Heroes ends on September 8. Assuming you take advantage of the many bounties available, you can easily earn enough to buy an ornament set for (at least) your favorite class before the event is over, even with mild play.
Of course, with the event bounties themselves currently awarding zero Bright Dust, that's a bit more challenging at the moment. Again, the best thing to do here is just hold onto your completed bounties until Bungie confirms that it's fixed the bug, which will hopefully be before the next weekly reset on August 18.
Here are the basics of the Destiny 2 Solstice of Heroes 2020 rules and rewards.