Are anime still submitted to broadcasters in tape form? In Shirobako, the studio has to deliver the tapes to broadcasters physically. I'm not surprised that this is a process, but I am surprised that there's nothing more modern in place. 

Early on, we learn that Musani Animation has an FTP server to store their files - So why don't they just send the file over the internet?



I've heard some arguments that physical deliver is still faster, but I'm not sure I buy that - Japan's internet speeds are quite good and the file sizes shouldn't be overly huge as it should be the compiled product rather than files with image layers, audio tracks, cg models, etc all seperate.

Are anime really still submitted by tape? If so - why?
No, tapes are no longer used. The tape appearing in Shirobako is just a joke in reference to old times. DVD-R are used nowadays, and maybe sometime in the future, blu-ray will replace them. It seems like the trend of using DVD-R began at least in 2007. On a side note, shirobako are not necessarily white either in reality.

As to possible reasons regarding why they have to give it physically:


They are late so they don't have the time to send it by post.
The FTP server was down in the anime so they had no other choice.
Mark of respect and seriousness for the broadcasters.
No problem of "timestamp" proof (if the broadcasters refused to accept the shirobako because of lateness) since a person goes directly to give it.
Reduced risk of leaking the episode if it was given by another mean.


Sources:


moto-neta.com/anime/shirobako/
detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1185176891 
img.animeanime.jp/imgs/thumb_h/44928.jpg
the anime Shirobako itself
Lots of other pages on Google when searching for "白箱"