Sorry, I don't know programming.
Everything you'll need (or whoever your programmer is) will likely be contained in the above link.
Now, to answer your questions: 1) yes there is a Twitter API that can be accessed; 2) the API for Twitter does allow users to post tweets directly from their own account in a different app.
Hope that helps!
Yes, there are several apps that allow you to Tweet directly, such as Hootsuite and Buffer, along with other enterprise-level solutions. Yes, there is a Twitter API. However, this particular API has proven to be difficult for a lot of developers and they have left the space.
If you're going to go this route, be sure you hire a developer who knows how to get their app working with the "new" API.