Partial submissions

Does jotform allow the admin to collect information on unfinished forms? 

Looking for something with conditional logic, partial submissions and section-based forms - that allows a user to look at a single page at a time, answer a few questions and hit "next"

It is possible to allow the users to save the form and continue later, but you will not be able to see the data they entered until they finish the form and submit it. 

Setting up the Continue Forms Later Feature in Card Forms

If you want to create an approval workflow form, please check this guide:

A General Approach on Building an Approval Process Workflow

I would suggest you to create a free account and test our Form Builder. You can signup here - https://www.jotform.com/signup/