Can my customers update their payment method information through the Chargebee customer portal?

Yes, this is possible.

Once they are in their portal account, they can update their Card details by selecting the 'Edit' option from under the Payment Method Information (Single Page)/ Payment Methods(Inapp) section in the portal. Screenshots below.

Customer Portal - HP V2(Single page)

Customer Portal - HP V3 (In App)

If you're not using the Customer Portal, you can also email customers with a secure link asking them to update their payment details. This can be done using the Request Payment Method option on the customer details page.

Here's more on that.