Toria,
Babies this age have little to not control over their extremities. And they are just in the initial stages of understanding "cause and effect". I'm not sure it's likely that she's attempting to make herself sick for your attention.
What might be more likely is that when you put her down, she doesn't know how to self soothe. Her hands are near, and they can be sucked on. Babies suck to self soothe. Fingers don't feel like a soft booby, and you are no where near her, and she gets sad and scared.
Just like we have to teach our kids to walk and talk, we have to teach them to self soothe. I wouldn't put baby in front of a book, and expect her to learn to read on her own, and I don't expect my babies to learn to self soothe on their own.
When I put baby down, I make sure for the first few minutes, there is something there to get their attention. Then I make sure I'm always talking and pop into her field of vision often. Babies don't understand that when you go away, you come back. If they can't see your face, they think you are gone. This is why peek-a-boo is so fun to play with them. They don't know you are still there when you hide your face! And they are so excited when you come back
Anyway take it slow, I know it's frustrating when we have things to do - but maybe you can wear her in a sling or carrier when you do these chores. I have fun things in different rooms that I can put baby in when I do chores. Or better yet, I let someone else who loves her play with her for me
Before you know it, she'll be toddling around and heading off to school. These days go by so fast!