Are anime still submitted to broadcasters in tape form?
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"