This turned out to be so ridiculously easy I probably shouldn’t even mention it, but only because I remembered that Update Manager exists. I thought I was going to have to burn an ISO, go to the datacenter, pop the disc in, etc. But with Update Manager this is all handled remotely. Just download the files from vmware.com and upload them to your vCenter server through the client. This YouTube video by VMware walks you through the procedure:
ESX to ESXi will have to wait for another time.