Many colleges and universities offer graduate degrees in reading, and many offer an online option. Other programs work around a teacher's schedule. Most universities realize that teachers pursuing a master's degree are already working in the classroom. They want to provide a program where you can further your education, but you still have time to teach your students.
For example, at the University of Missouri Columbia, reading classes are offered starting at 4:00 pm or later on weekdays, after most schools are out. Some classes can be taken as an independent study, where you work one-on-one with the instructor and meet with her to discuss your progress when it is convenient for your schedule. There are also several classes offered in the summer, when teachers are out of school. Some of the classes are offered online, so that you can do a combination of both in-person classes and online.
Another example of a program is at Capella University in Minneapolis, Minnesota. This is an online master's degree in reading program, where you can receive a master's degree in reading and literacy if you are already a certified K-12 teacher. For more information on other graduate schools with online programs, you can check out the website, GradSchools.com. You can do a search by program type and/or location to find one that best suits your needs.
If you do not have a graduate degree in reading or a graduate degree in education with an emphasis in reading education, it will not be possible to have a career in education as a reading teacher. Instructing students, especially ones struggling with reading, takes special training and knowledge.