My apologies that you didn't get a reply here. We're handling this in the support queue, but for anyone who comes along:
You need to create a CampaignMember object for them. In order to do this you will need to know:
1. The Contact/Lead Id for the respondent.
2. The CampaignId for the Campaign you wish to add the respondent to.
This is a bit hard to talk about in the abstract, so if you could tell me which form you're working on, and provide me with the CampaignId, I'd be able to help you better.
I'm trying to figure out the same thing but with a new lead. Is that possible where we won't know the lead ID because it will be created when the form is submitted?
Yes, just reference the new record in the connector mapping:
Salesforce field: Contact Id ---> 'get value from' 'id of object above' [#1: Contact]
That will work even if the Contact is being created. More details are in the documentation for the Salesforce connector, and if you have any problems, open a support request and we'll help further.