Students select majors without the optimum background all the time. Physics 1 is offered for a reason. Not all colleges require an applicant to show interest specifically in engineering, aside from demonstrating an aptitude in math and science. Of course it is better to have a physics class under your belt. I would definitely recommend taking an online class during senior year if you can fit it in.
Also, @eyemgh made an excellent suggestion to consider Industrial Engineering. There can be a lot of overlap with business and it tends to be lighter in physics than some other engineering majors (though physics is still a requirement).
It is important to consider what you would like to study. If you aren’t sure, it’s fine to be open to engineering. But, it would be a really poor choice if you have little interest in engineering. At a minimum, I would suggest attending a school with many options.