To make the process as smooth as possible, so it does not affect your account or delay inspections here is how:

Step 1: First log into your SnapInspect 2 administrator account here.
 
 Step 2: Once logged in, at the top right corner of your screen you will see "Settings" click this field.

Step 3: You will see multiple tabs on the left side of your screen, click on "Account & Billings."

Step 4: Click on the "Click here to update your Credit Card" button

Step 5: A pop-up window will come up, at this point you are ready to enter your new payment details
 
 

TIP - If you are trying to update your SnapInspect 3 payment details, this will look somewhat different, here is the guide for that process here.