No, the API only allows you to connect to existing Stack Exchange sites, and Stack Exchange no longer sells public-facing sites. Stack Overflow Enterprise is available for internal use, but it is really meant for larger organizations only.
It looks like you want one of the clones listed on Metathe clones listed on Meta.
Since it's not clear which, if any, of those clones offer an API, there is another approach...
If you submit your site through the Area 51 proposal process and it goes to beta, then you can access the new site through the API. It won't be private, but you could potentially get a lot more people helping as well.