The last genuine Mac edition video cards for the classic Mac Pro were the AMD Radeon HD 7950 (Mac Edition) and the Nvidia GTX-680 (Mac Edition).
It has been possible for people to DIY 'flash' PC edition AMD Radeon HD 7950 and 7970, AMD R9 280 and AMD R9 280x cards to add Mac firmware.
It is possible to buy various newer Nvidia cards converted and flashed to Mac firmware only from http://www.macvidcards.com/
There are some features that can only be used if the video card has Mac firmware, for example using the Option key to select alternate boot drives, accessing the RecoveryHD partition or Internet Recovery, or FileVault2 logins. However due to the fact that Apple do list the AMD RX 580 as a suitable video card it would seem clear that you do not need a video card with Mac firmware in order to run Mojave on a classic Mac Pro. (Doing the actual install might be a different matter.)
There is no way currently to flash an AMD RX 580 for Mac firmware.
My hope is that if Apple's forthcoming Mac Pro 2019 includes PCIe slots that this might encourage AMD and/or Nvidia to produce new official video cards with Mac firmware.