Hey @Sversave Welcome to the community :)

I believe you should be able to do this without using the Zapier integration. Our dearly departed @Francois (Typeform) (not dead, just recently moved to another company) put together a short guide to adding survey responses to a contact in Salesforce. We've yet to publish it on our Help Center but here's a heads up of his draft tutorial which gives all the details.
 

BTW I recently spoke to a property firm in the UK who have built a very cool multi-touchpoint feedback solution using Typeform and the native Salesforce app integration. They are hopefully gonna be sharing what they did here in the community in the next few days. I'll ping you in here as and when they post it :)

Hope this helps!