Does Soapbox Mailer update Contact records to show campaign activity?

Yep!  Okay, well, kinda.  Soapbox Mailer uses the standard Campaign Member object to make a relationship between a Campaign record (your Soapbox Mailer email) and a Contact (the recipient).  When you add a Campaign Member to a Soapbox Mailer Campaign, you're making a relationship between the Campaign and the Contact.  That means that on the Contact record, you can look in the Campaign History related list and see the Soapbox Mailer Campaigns this Contact has been associated.

Happy emailing!