Formal:
* Permit: to allow something to happen
* Authorize: to give official permission for something
* Sanction: to approve or give formal permission
* Condone: to accept or tolerate something that is morally wrong
Informal:
* Let: to allow something to happen
* Agree: to accept something or someone
* Tolerate: to put up with something even if you dislike it
* Endorse: to publicly support or approve of something
More specific:
* Acknowledge: to recognize or admit the existence of something
* Embrace: to accept something enthusiastically
* Welcome: to receive something or someone with pleasure
* Grant: to give someone something they have asked for
The best word to use will depend on the specific context. For example, if you're talking about a government granting permission for a project, "authorize" might be the most appropriate word. If you're talking about someone tolerating a noisy neighbor, "tolerate" might be a better choice.