We are looking into using the Customer Center. Is there a way to get the current customer that is logged in?

Use the N/runtime module and the function runtime.getCurrentUser() will return the user currently logged in. https://suiteanswers.custhelp.com/app/answers/detail/a_id/45166/loc/en_US

Note that in the customer center (assuming I'm remember correctly),

currentUser.id

will give you the customer's ID (i.e. from the company); if you're looking for the ID of the specific contact that's logged in, I believe you can get that from

currentUser.contact

.