I am trying to create a custom form on my organization's webpage, NOT a salesforce cloud landing page. How do I get this form to POST up to a SFMC data extension. Smart Capture Forms seem to only work if being used on a cloud landing page, but for my purposes I need the form to be native to our organizations webpage. Any help would be appreciated.
There are a couple of different options.
You could use Email Studio Web Collect. https://help.salesforce.com/articleView?id=mc_es_email_studio_web_collect.htm&type=5
It's basic and and has a number of limitations.
Alternatively for a more robust options that includes more control over updates you can use the REST API. https://developer.salesforce.com/docs/atlas.en-us.noversion.mc-apis.meta/mc-apis/postDataExtensionRowsetByKey.htm
There is also a SOAP API that you could also use. https://developer.salesforce.com/docs/atlas.en-us.noversion.mc-apis.meta/mc-apis/web_service_guide.htm
Another slightly more creative method could be to create a CloudPage code resource containing AMPScript or SSJS and have your website POST to that endpoint.