Can I get Invisalign if I need other work done on my teeth?

It depends on what work is in your treatment plan.  Rule of thumb is all active disease processes in the mouth should be treated and restored prior to orthodontic movement.

Yes, you can get other work done while getting Invisalign. However, it is always better to treat the active problem first. That means, in case any root canals are to be done, they should be done earlier. Any cavities that need fillings should also be addressed before. However, in case crowns are needed, and they can wait, should be done after Invisalign. Oral prophylaxis can be done during the Invisalign treatment at any time.
Good luck!