Depending on the age of the person, crossbites are usually treated with a palate expander. The expander can correct the problem and the person may not even need further orthodontic work.
Invisalign will not help a crossbite. Braces would be the better option to correct a crossbite. Braces will also be a cheaper alternative with great results.
No, a cross bite requires major tooth movement in order to restore proper occlusion and is not treatable with invisalign. Invisalign is designed for simple torsion and simple alignment of teeth. Instead, traditional brackets must be used. An orthodontist can help with diagnosis and treatment planning of your dental needs.
Invisalign purportedly helps the symptoms of a minor cross bit, but it will not really work as a long-term solution. You may want to talk to your dentist about braces.