Working out at home is totally fine! The problem a lot of people have is they start way too big. Start at home like one or two days a week and work up when you’re comfortable with it. Hell you don’t HAVE to go to the gym. There’s loads of stuff you can do with just your body weight and even more with a simple set of dumbells + smth to sit/lay on like a piano bench.
I also only work out at home. But also be careful sometimes if you drink too much water at once after heavy exercise it can also make you throw up (learned that the hard way) take little drinks of something with electrolytes (gatorade etc) frequently instead of gulping water all at once. Just in case that’s also causing some of the vomiting other than fear