Of course you can!
I think it is a myth that you need to read a lot to be a good writer. I have never seen a scientific study that supported that claim.
Do you need to watch a lot of sports to be a good sportsman? Do you need to listen to a lot of music to be a good singer? No. It will certainly help in your understanding of a craft to witness a master at work, but I don't think observing a craft has a similar impact on your performance as actually training that craft.
You certainly need to be or become a master of language, but language does not only happen in literature, and much of contemporary literatur is not even written in a "literary" style, but rather emulates oral history and audiovisual media. And those that read a lot don't usually write a masterpiece on the first try, either. They, too, have to learn how to actively use the language they have until then only consumed passively. Because, as we should all know, eating a lot is not the same as understanding how the food is made.
Just do what you like and don't worry about dogmas.
Also, let us look more closely at the exact claim that is made. Because it is not said that writers simply need to read a lot, but that they need to read literary works.
Everyone who can read and write has read something. It would seem improbably that someone knows how to write without being able to read as well, so you have read texts (in the widest sense of the word) and you very probably still read more than you are aware of even today. For example, it is commonly claimed that girls read more than boys. In a study (which I am too lazy to find now) the authors found that boys do in fact read just as much as girls, only they don't read books! What they read is: magazines, cereal boxes (sic) and non-fiction books. But because "reading" means "reading fiction novels" in the minds of most parents, teachers and scholars, it seems as if boys did not read as much as girls.
Also, as I wrote, language is not only written language. There is spoken language as well. Some of the "literary" masterpieces where composed without writing, purely in the minds of their authors, and "published" by orally narrating the work to their audience. Most ancient and medieval literature is of this type, and it was often not recorded by the author, but by someone who heard the narrative, and often not directly from the author either. So, even if you assume that the ability of composing narratives requires familiarity with other narratives, you can become familiar with narrative traditions and standards by listening to audio books, watching movies, and playing computer games. After all, writers use books that teach screen writing to learn writing novels! Why should watching movies not help them understand plot structure, characterization, and current motifs, then?
Finally, reading a lot of something will not make you have more ideas. Much of genre fiction is just plagiarism: having the "idea" of writing a story like what you have read. Tolkien did not read Fantasy to write Fantasy. Wells did not read Science Fiction to write Science Fiction. The greatest works establish new genres, and their authors have never read anything like what they themselves wrote.
Finally, assuming that language and style can be learnt by other means than reading fiction, there are many writers whose first books was not written because they loved to read, but because they had had experiences or gained knowledge that they wanted to impart. Think of all the autobiographies. Of course some autobiographists loved to read and had read a lot of books before they wrote about their lives. But some have not, and the first book they read was their own.
And since writing is not only about literary quality, but also about writing something that interests your readers, the claim that books written by writers that did not read a lot are less elegant and eloquent does not really matter, either.
As a final thought, though, I am quite sure that writing will make you interested in reading.
