Formal:
* Authorize: This implies formal approval or granting of permission.
* Sanction: This suggests official approval or acceptance.
* License: This usually refers to a legal document granting permission to do something.
* Empower: This suggests giving someone the authority to do something.
* Allow: This is a more general term, simply meaning to make something possible.
Informal:
* Let: This is a casual way of saying "permit."
* Approve: This is a more informal synonym for "authorize."
* OK: This is a very informal way of saying "permit."
Other:
* Enable: This implies making something possible or easier.
* Facilitate: This means to make something easier or smoother.
The best synonym to use will depend on the specific context of your sentence.