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, you can integrate with Odoo Cloud (Odoo Online) using their API, but there are a few things to keep in mind.

Odoo Online does support the standard XML-RPC and JSON-RPC APIs, so you can connect your existing C2C marketplace to it for things like CRM, payments, and invoicing. You’ll be able to push customer data, create leads, generate invoices, and even trigger payments if you're using supported payment gateways like Stripe or PayPal.

The main limitation with Odoo Online is that you can’t install custom modules. So if your integration needs custom fields or business logic, that could be a blocker. In that case, you might want to look into Odoo.sh instead — it's still cloud-hosted but lets you customize more freely.

If your needs are mostly standard (like syncing orders, invoices, and customer info), then yes, the API is usable and works pretty well for that. Just make sure to create an API user and authenticate via their JSON/XML-RPC endpoints.

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