Permission/Authorization:
* Permits
* Enables
* Grants
* Authorizes
* Sanctions
* Approves
* Lets
* Gives permission
* Gives the green light
Possibility/Opportunity:
* Makes possible
* Facilitates
* Enables
* Permits
* Offers
* Provides
Tolerance/Acceptance:
* Accepts
* Tolerates
* Endorses
* Concedes
* Suffers (implies begrudging acceptance)
To Give Space or Freedom:
* Lets
* Gives room for
* Provides space for
Other:
* Admits (used when referring to physical entry)
* Provides (used when referring to something tangible)
To choose the best synonym, consider the specific context in which "allows" is used.