Creating a two part application that requires approval before the user can continue the form

I would like a application form that allows the user to fill out half the form, submits it, then it will email notify me, from there I can approve on deny it, and if I approve it, a link will be sent to the applicant to complete the second half of the form.

Yes, that's possible. You can hide fields in your form and only show them after your approval. You can also fire out email notifications depending on your approval selection, either approved or declined.

First, here's how to configure approve/reject submissions. Guide: https://www.jotform.com/help/194-How-To-Assign-Status-To-Submissions

Through this guide, you can add Show/Hide conditions that IF "Approved" is selected, you can show the rest of the fields for the users to fill. You can also add an email condition for the same IF rule that sends an email notification to the user informing them about the status of their submission.

Guide for email condition: How to send to a specific email address based on a selection

Guide for show/hide fields condition: How to Show/Hide a Field

Here's an example form I've created for you https://form.jotform.com/201738706872058. Feel free to clone it so you can inspect how it is configured specially the email and show fields condition.

If you need further assistance, let us know.