"I don't think it's even possible to purchase any insurance policy strictly by phone."

Call Geico, give them the Plates, VIN, DL#, and some personal info, a Visa and you get a binder faxed/emailed to you in less than an hour. A week or 2 later you get a signature pack in the mail to return to them. Leagally insured to drive in an hour over the phone, an accident the next day would be covered without the signature on the return mailer.
Well, that's what I did in the independant country of Texas(currently a US state) 5 months ago. State laws vary.