I kept studying to the point that I became dizzy.
Can that be switched around to become this and still be grammatically correct?
To the point that I became dizzy I kept studying.
Is there anything wrong with that sentence?
As several people have said, the original (with a comma) is perfectly grammatical. However, when you switch it around, it sounds better with "kept on".
To the point that I became dizzy, I kept on studying.
and I'd be more likely to reword the switched-around version with an "until"
Until I became dizzy, I kept on studying.
or, maybe keeping closer to the meaning of the original,
Until it made me dizzy, I kept on studying.
It needs a comma after dizzy.
For me, it doesn't work as well that way around because you break the intimate connection of "studying to the point ..."
It is grammatical, though probably needs a comma, a RedGrittyBrick says.
It is an example of topicalisation, and puts a strong emphasis on whatever you've put to the front of the sentence.