I have a field on my donation form for the Membership ID but I do not see any documentation on how to pass that information through an API call.
Luminate Online does not automatically populate the membership_ids on constituents' profile. These values would need to be manually populated.
However, if you are making an API call and already have a custom field "membership id" on the donation form, you can add a parameter for "membership_id" that would pass a value to that field on your constituent's profile.
You can find reference material for this in the section labeled "Custom Fields" in our API documentation on donation API calls.