You should not need a SSN to enroll your newborn in your health insurance provider. Check with your insurer for their exact method of handling this, but in general, you can add a newborn without a SSN.
For example, from this article by a major benefits company, they note:
If you're a new parent who needs to enroll a newborn within 27 days of their birth (a qualifying life event), you don't need to provide an SSN when adding your newborn to your existing coverage. Newborns often don't receive an SSN until several weeks after their birth, but must be enrolled as a dependent within 27 days of birth.
They go on to suggest that if a SSN is "required" by the page, to put in a dummy value (like 111-22-3333), and then update it later with the insurer; I would not necessarily suggest doing that unless your insurer tells you to, but they should be able to tell you what the specific right answer is for them (or perhaps add the newborn for you on the phone).
To directly answer the question as asked, though, unfortunately going into the office is the answer. As this blog post explains, temporary cards are available, but only in the office - otherwise you have to wait for the mail!
