I have one friend who was a single mom that worked full time & homeschooled 2 children without issue & both went to college/uni after & she is what made me consider it in the first place. I saw how great her kids were.
I don't think it's a real stretch to think you can do it if that is your real desire. My friend was truly a single mother during most of the year, and she managed.
I homeschool (I am not a single mom, however. Your situation is different.) We take no major breaks. If we go visit family on vacation we still do school, maybe just less, like an hour a day, etc. We often do things on weekends too, and we go all year around. It advances them in grade levels faster & it also allows you to create an environment where learning is the norm.
I could for sure see me working full time & being able to do it. It is helpful to be home in the day so you have the option to belong to homeschool groups. Most of your activities, meetups & coops are going to be done in day hours. So if it were possible, I'd opt for a job that left as much of the day open as I could so that I could tap into that community & support. It helps. My friend did that. She went into work at something like 2-3pm (I can't recall exactly.) Her kids are grown & have kids now.
Filing paperwork will depend on where you live. Where I live I file nothing. When I was first interested I didn't let myself get deterred by all the people that said it would be difficult, and am glad of it. It's not super easy, but I have friends with kids the same age & the stuff they do isn't super easy either (think homework, class projects, teasing, bullying, being left out, etc.) School issues can be difficult.
I was scared to start, but it has done nothing but make our lives easier in the long run & I would continue it one way or another if I found I had to go back out & get a job.