Is it Islamically permissible to use braces on the teeth or to get an Invisalign, which is a clear dental brace for teeth that helps to align teeth and close the gap between the two front teeth?

The use of braces due to need, such as setting right a defect or straightening crooked and unaligned teeth, is permitted. However, using braces purely for extraordinary beautification purposes is to be avoided.
A person can have his teeth treated, or have them straightened if they are crooked, just as he is not prevented from treating any other sicknesses within his body.
What matters is that he should not do that merely for the purpose of beautification, because the general principle with regard to cosmetic procedures is that if they are done to remove a deformity or fault, there is nothing wrong with them, but if they are done merely for purposes of beautification, then they are not allowed.