I am trying to gather project ideas in a Sharepoint List that users will fill out via a web form. Does anyone know how to connect the two, so after the user clicks Submit, the data will appear in the SharePoint List?
By web form if you are referring to a .Net page, you could use CSOM to connect to your site and create a new list item. MSDN article on CSOM
Another great idea would be to use REST API if you cannot have managed code.
Research a bit and you will be able to make a choice.
If you're hosting your own page as a form, you could use the REST-API from the server. This would enable you to use a techical user to access SharePoint. REST-APIs are usable from any language (PHP, Java, etc.)
If you're hosting your own page as a form on an asp.net-enabled server you could also use CSOM. Again, this would enable you to use a techical user to access SharePoint.
You could of cause host your form in SharePoint and use SP.js to access the List-data. If you'd do that (hosting in SharePoint) it would probably be better do directly use the NewForm.aspx.