I'm not entirely sure exactly when this started working, as I never really heard or saw any 'official' announcement of it (as I seem to remember happening when Tumblr + Soundcloud and Tumblr + Spotify became available). Which is a bit surprising, at least for me, as I have been wanting to be able to use Bandcamp URLs for Tumblr audio posts ever since Tumblr first started their external URL feature as an alternative to uploading your own tracks.
I have been a bit preoccupied with ‘away from internet’ things these past couple months, so perhaps I just missed the announcement and everybody already knows that this is now possible. But if not, here it is in picture form…
But it was all new news to me when I stumbled across it being mentioned on Ben Walkers tumblr (and since he has insider Bandcamp info, I'm going to guess this integration started working around the time of that post), and when I tried googling to see if there was a more official announcement, I never found one. Although I have seen a few people already using it, so I guess the cat is definitely out of the bag already.
It doesn't look like you can just 'search' for tracks that are available on Bandcamp (as you can with Soundcloud and Spotify), but I never use that feature anyways, so as far as my Tumblr + Bandcamp integration needs are concerned, I'm all good.
Anyways, I just thought I would also share this info in case there are any other Tumblr users out there who, like me, didn't know how easily you can now spread tasty Bandcamp tracks on your blog.