Yes. an example of this of an English Book becoming an anime is Deltora Quest
The Deltora Quest series is the collective title for three distinct series of children's fantasy books, written by Australian author Emily Rodda. It follows the adventures of three companions as they journey across the fictitious land of Deltora, endeavoring to recover the seven gems stolen from the magical Belt of Deltora and defeat allies of the evil Shadow Lord.
under Adaptations it says
A 65 part Deltora Quest anime series of the first eight books began its broadcast season in Japan on January 6, 2007. Rodda chose this option because she and her kids "love Japanese anime, and want any adaptation of Deltora to be cool".
this is backed up by it being on the anime's entry on Wikipedia under Production
The series was produced by Genco and the animation production at OLM and SKY Perfect Well Think. Rodda was originally approached with many film offers, but it was only this studio that promised to not change the story. The first episode was aired on January 6, 2007 in Japan.
Deltora Quest was originally licensed in North America by Geneon as one of their final licenses, but shut down before doing anything with the series. The series was then rescued by Dentsu's newly established North American branch, and they produced an English version of the series in association with Ocean Productions in Vancouver, British Columbia and recorded at their Calgary-based Blue Water Studios.
and since the anime only started being aired in the west in May 2010 this is an instance of an english written book being made into a japanese anime. while not American Emily Rodda is Australian and thus English
however making an "english anime" and dubbing it in japanese we have 2 close examples to this
The series is presented in a style that combines anime with American cartoons and relies on the imagery of East-and-South Asian, Inuit, and New World societies.
however i don't know if it was ever subbed in Japan, though if not the possibility that it could be exists as Japan does localize western movies like Toy Story (it all depends on the licensing). i included this because fan still debate if this is anime or not where it was created in english
a better example however is RWBY
RWBY (/ˈruːbi/, like "ruby") is an American anime-style web series and media franchise created by Monty Oum for Rooster Teeth.
unlike Avatar we know it's been localized in Japan having been dubbed in Japanese
The series has also been dubbed in Japan and broadcast by Tokyo MX, in partnership with Warner Bros. Japan.
and also it's spawned a manga too which unlike Avatar's comics, was produced in Japan
The November 2015 issue of Shueisha's Ultra Jump magazine announced that Dogs manga author Shirow Miwa would be illustrating a manga adaptation of RWBY, which debuted in the December 2015 issue on November 19, 2015
however as кяαzєя pointed out this is rare. aside from the costs of the license to reproduce and distribute dubbing also increases the costs over subbing because of hiring Seiyuu and as we see with RWBY it had the backing of major companies like Wanner Brothers who do operate in Japan to (you'll see alot of Disney stuff gets localized because Disney operates in Japan too)
however, just like кяαzєя said there is always the doujinshi route. admittedly in the west (to my knowledge) it's not entirely the same where there are large conventions centered around presenting and selling doujinshi however as doujinshi (manga and light novel wise) are really just self-published work and self-publishing books has been made easier1 with the internet and digital distribution
1: some publishers like Amazon which facilitated self-published works have been trying to regain the control traditional publishes had since with self-publishing, traditional publishers doesn't have nearly as much control over gaining profits and intellectual propriety rights as they used to