Formal:
* Authorize: To give official permission for something.
* Sanction: To give approval or permission for something.
* Empower: To give someone the authority to do something.
* License: To give official permission to do something.
* Grant: To give something, especially formally or officially.
* Approve: To officially accept something.
* Allow: To make something possible.
* Enable: To make something possible.
* Condone: To accept something that is morally wrong.
Informal:
* Let: To give permission.
* Allow: To make something possible.
* OK: To give permission.
* Go ahead: To give permission.
* Give the green light: To give permission to start something.
Other related words:
* Tolerance: The willingness to accept different views or behaviors.
* Acceptability: The quality of being acceptable.
The best synonym for "permit" will depend on the specific context in which it is being used.