I attempted to set up a legacy contact, and when I got to the end of the process, a dialogue box popped up and said try again later

If you already tried (to no avail) ALL the steps provided in the support article that led you here (specifically the section "What you need to add a Legacy Contact for your Apple ID"), AND if you already checked that Apple Servers are okay (using this support page:--> Apple - Support - System Status), AND if the issue still persists, I suggest that you contact Apple Support directly, using this link:--> Get Support

Thanks and regards

Giulio