Send data to client's CRM from embedded marketo form

I'm assuming this is achieved via API.

Don't assume.

Or, more precisely, don't assume that "API" is a meaningful enough term. There are many different types of API, and the mere existence of something branded as an "API" doesn't mean the specific function you need is offered via that API!

At any rate, the question is fundamentally about their CRM's capabilities, not about Marketo's. We know how to get data into Marketo.  If their CRM has a forms-compatible API endpoint, i.e. an endpoint that is built to accept form posts, we can cross-post there from the browser when the Marketo form is successfully submitted. (Alternately, you can do that same thing via a Marketo webhook, but posting from the browser is vastly preferred for scale and stability.) If their CRM doesn't expose such an endpoint, we need to know exactly what it does expose.