As someone with a Computer Science (CS) degree and who has worked in the field for nearly three decades, I expect that you will have no problem getting a software development job with an EE degree. On occasion, I have worked side-by-side with people who have EE degrees instead of CS or Software Engineering (SE) degrees.
An EE degree has been preferred in some circumstances: I've seen this when the work is particulary close to the hardware, or in some settings where CS and SE are not seen as "real" technical disciplines, but EE is accepted.
While it's not the case so much anymore, when I got my degree 2+ decades ago many schools didn't have majors in Computer Science. At those schools, people who wanted to become software developers most often majored in EE or Mathematics. Futhermore, I don't remember seeing a Software or Computer Engineering degree offered anywhere until well after I had completed my Bachelors.
In point of fact, software development is often one of the more flexible fields when it comes to what degrees can get jobs, or even if a degree is needed. I've usually (but not always) worked in scientific environments and have had programming co-workers and bosses who had degrees at varying levels in CS, SE, EE, Mathematics, Astronomy, Physics, Chemistry, Geology, Meteorology, Oceanography, Business, and even no (college) degree at all. I also have friends who work professionally in software development who have degrees in other fields such as Mechanical Engineering, Chemical Engineering, Psychology, Philosophy, Education, History and Religion.
In observing requirements in want ads, it does seem that things are changing somewhat - more software development job listings want a degree in Computer Science, Software Engineering or a related field. However, (in my mind at least) Electrical Engineering is a related field and I do not think that is likely to change in your lifetime.
As for how to go about getting a software development job with an EE degree: While in school, take as many courses as you can that teach software development (if you can, get a Minor in CS/SE/CE/whatever your school offers). Also, you can do some open source development. These things will build a portfolio and resume that should help you get interviews when you apply for jobs. Once out of school, I recommend that you apply to jobs just as if you had a CS degree. Once you have that first job, it should be even easier to get other jobs in software development.