How can I integrate the OpenStreetMap directions on my web page, Is there an API?

OpenStreetMap itself does not have, to the best of my knowledge, a routing server. It seems to rely on other projects (OSRM, GraphHopper, etc.). Those projects may have an API and you may want to check on their websites to see what they have to offer.

If you don’t need things tightly coupled and/or using your own visual look and feel, then you can do some stuff without a API.

For example, a screen capture of your business location put into a web page with the image being a link to that view on OSM's main page. Or a link for directions that goes to a routing project's page with the to and from fields already filled out.