After not playing Overwatch for some time, I installed it through Steam. After linking my Steam account with my Overwatch account, I was able to access all of the heroes (as expected), but none of my cosmetics (skins, golden guns, etc.)
I contacted Blizzard Support, and a support team member said they were able to see the cosmetics on my Battle.net account and refreshed it, which should have made my cosmetics appear on my Steam. Since it didn’t, they told me to file a bug report here.