The issue may that the field you are mapping to is looking for the
related CONTACT (based on numeric value), rather than the just the contact name (text). You can test to see if this is the case by adding a text field (not a lookup) and seeing if you can map the contact to that field. If it is a field that you cannot edit on a form, its likely you will have issues mapping directly to it using Zapier. If you can confirm we can likely come up with some workarounds.