The Odoo Online has API.

You can see that documentation has a note about Odoo Online instances (yourdomain.odoo.com):

https://www.odoo.com/documentation/17.0/developer/reference/external_api.html

UPDATE:

It depends on your plan (as Chris TRINGHAM already said):

https://www.odoo.com/pricing-plan

Without API it is useless. Many services connects via API.

Odoo is too expensive if you want to go live with it.

A cheeper way is to have it installed locally for your company.

And have a live website like wordpress on a shared hosting.

Then connect odoo and wordpress via wordpress API.

PS: Later you can replace wordpress with custom PHP scripts that connects to different services. 

PS2: From your website to odoo you need to get just new customers and new orders. (I don't know why people ask to link all data and then to pay hundreds for such complex  API integration).

From odoo to your website you may need just to update the order status.

So for all services and visitors - live website.

For accounting - odoo (installed locally for you company).

Stanislav

Yes, but you need to pay for the "Custom" version of Odoo Online, not the "Standard"