There are some engineering management programs that require that you have experience, but many do not. In fact, some programs, like Duke's Distance Master's of Engineering Management would like students to have fewer than five years work experience and have tailored the program to individuals early in their careers. Meanwhile, at the University of Texas at Austin's Online Master's in Engineering Management, they would like students to have at least 18 months of work experience and Drexel's Online MS in Engineering Management wants at least five years for their program, though this is not required. Oklahoma State University's Online Master's of Science in Engineering and Technology Management, meanwhile, requires "relevant and current employment since graduation with a bachelor's degree."
So, check in with the programs you are interested in pursuing. Do your research. Speak to engineers and managers in the field. Ask professors or alumni from engineering management programs you are interested in what they would suggest.
Either way, there are pros and cons to taking time off and gaining work experience versus jumping right into an online master's program in engineering management.
Jumping straight from a bachelor's to a master's in engineering management program means that you are still in "student mode." You are used to living a student lifestyle and you remember how to be a student. After a few years away, those skills get rusty and it can be a challenge to get back up to speed. Also, if you are interested in pursuing a research focus, staying in the academic world can be useful in providing you with opportunities to contribute to a body of research.
While work experience is likely not necessary for acceptance to a program, there are certainly reasons to take the time off to gain some before beginning graduate school. Having work experience can also open doors for research work while you are in graduate school. This experience makes you valuable to researchers because it helps you understand the true challenges in the field, not just the theories. Another reason to consider gaining work experience first is that it gives you the benefit of making some money before continuing with school, and as engineering is a lucrative field, it can truly help you plan financially for your graduate work. And one last bonus of having at least a few years of experience? In some cases, you might be able avoid having to take the GRE or GMAT for entrance to the program.
Gaining work experience before your master's also helps you determine which path you want to pursue. There are so many different engineering careers, take the time to find what suits your interests and skills. Also, not all engineers desire to move into management, and pursuing a degree in engineering management will put you on that track. If you aren't sure if management is something you're interested in, take time working as an engineer and observe what managerial roles look like at your organization. If during your time at your organization you find there are aspects of management that are appealing and fit your skill set or career trajectory, then a master's in engineering management might be a great choice for you.