The term "primary owner" suggests that one of the joint owners has a preferred status in some way, and that is, of course, not the case. It is, though, typically the case that the first-named of joint owners is the party whose tax identification number is used for interest reporting to the IRS. From your question, I assume that the joint ownership was with rights of survivorship. If that is correct, ownership of the account vests by contract and/or by operation of law in the surviving joint owner immediately upon the death of one of the owners. There is certainly no need to obtain a new signature card, although it may be appropriate to simply note the death of one of the owners on the current card.
If you did not obtain a W-9 certifying the TIN of the surviving owner at account opening, do so now, and update the record so that tax reporting will be done under his or her TIN. While doing that, it's appropriate to remove the decedent's name from the account title and make any needed change to the mailing address.