The short answer to this question is yes, of course, you can be a writer without doing a lot of reading. But the complete answer is that you will be a better writer if you are also a reader.
Think of it this way: can you be an athlete without practicing? Sure. You can get by (even fairly well), if you’re blessed with talent. But you can never be the athlete you could be unless you eat well, train, stretch, rest, and learn everything you can about the sport you play.
If you are or want to be a writer, reading should be part of your practice schedule. There is great learning that takes place when you read. Your mind is naturally drawn to certain writers, because you like their style, their voice. Reading what they write makes you want to write, and can put you in the mood to start writing before you even finish the reading.
Think about what you enjoy reading most, because it can often be an indicator of what you should probably write. In other words, if you subscribe to In-Fisherman and devour the meaty how-to stuff, that might be a good place for you as a writer. If you subscribe to Sporting Classics and love getting lost in good adventure writing, that could be your place.
A perfect world would have you reading what you love to read, and writing what you would love to read. Follow that path as much as circumstances allow, but as a practical matter, writers who come to depend on paychecks from it have to pay attention to what sells best, too. In order to write well outside your primary areas of interest, reading others who excel in popular categories can be a difference maker.
Reading is fuel for your writing. It’s hard to go to the well of words day after day without pouring a few gallons of fresh treatments in the tank. Find writers that you like. Read what they have to say. Your own work will be better for it.