A site owner or user with full control should / would normally be able to create a sub site.
However sometimes we get calls where by a site owner cannot create a sub site, when they submit the create new site form they get an access denied page display.
If you are getting this error check the Taxonomy hidden list permissions on the site collection in question:
In our case no one had any permissions assigned to the list which was causing the error. By default ‘Everyone’ should have read access to this list. Once read access was granted back to the list, site owners could once again create sub sites.
This list stores copies of any terms that are used in the site collection from the managed metadata service. Search for this in a search engine if you want to read up more about this list & why it exists etc.