Are anime style drawings acceptable?

Any style is allowed. Since you have to submit three pieces, a mix of styles is probably even better, since it shows range.

The more important question isn't your style, it's this: does your art tell a story? Portraits or landscapes don't do as well as action scenes where something is happening and the viewer really wants to know what happens next. You can certainly do that with anime.

I can recall a few recent illustrations that I'd call very 'manga' influenced, without being full-out (I'm not very informed on the nuances) ... I'd say they were manga-influenced but not as "cartoony" as most I've usually seen?

I'd echo Martin: show range and depth. Manga per se is rather stylized, rather codified, isn't it? There's a skill set to drawing BARBIEtm comics too, but ... I'd suggest the contest is looking for "illustration" more than "comics" in a broad sense. Do the rules say they don't want a page of comics art, i forget ...

I can recall a few recent illustrations that I'd call very 'manga' influenced, without being full-out (I'm not very informed on the nuances) ... I'd say they were manga-influenced but not as "cartoony" as most I've usually seen?

I'd echo Martin: show range and depth. Manga per se is rather stylized, rather codified, isn't it? There's a skill set to drawing BARBIEtm comics too, but ... I'd suggest the contest is looking for "illustration" more than "comics" in a broad sense. Do the rules say they don't want a page of comics art, i forget ...

Brad's "Exanastasis" had a very manga-esque illustration. One of the best in the book (#26), as I recall.

Anime is a type of art that is used in Japanese cartoons. Is this style of artwork allowed in this contest?

http://writersofthefuture.com/past-winners-galleries

There's a useful gallery of some of the past contest winners. There are definitely anime-inspired winners, although I don't think there are any that stick strictly to the tones, proportions, and styles used by (most) contemporary manga and anime. (It's worth noting in passing that "manga" only means "comic" and "anime" only means "cartoon." There's as much variation in illustrated stories published inside Japan as there is in the differences between Marvel, Dark Horse, and uh... whoever else it is that publishes comics here in the US.)