If your account is part of an organizational license, reach out to your brand admin; that person can coordinate with Qualtrics.
If you have an individual account, reach out to your Qualtrics account exec (if you have one) or Qualtrics support for further assistance. More info is at https://www.qualtrics.com/support/survey-platform/sp-administration/brand-customization-services/vanity-urls/

First of all, thank you, Matthew.
So, In conclusion, do I need to work with Qualtrics to set up both requests?

I can't say for sure without knowing more about your Qualtrics setup and the specifics of the client's project(s). Does this client have just one survey project they've acquired the vanity URL for, or will they have several projects over time? For the client's current survey, is the survey link going to be a single, anonymous URL that every respondent uses, or are you distributing Individual links directly to each respondent (by email or some other distribution method).
If all respondents are accessing the survey through the same anonymous URL, then the client could set up a redirect from their vanity URL to the Qualtrics survey URL. But if the client is looking for a Individual links that are unique for each respondent using the vanity URL as the root, then that has to be done through Qualtrics, and may only be available with certain types of accounts.
If I understand the second request properly, the client wants a specific URL for the survey (beyond just the root/vanity part)? If that is correct, to my knowledge that is not possible in Qualtrics. Users cannot control the full URLs that Qualtrics generates. But, as mentioned above, if the client is using a single anonymous URL for the survey, then a redirect could be set up from any URL in the vanity domain without needing to work through Qualtrics.

I guess that we should redirect from their vanity URL to the Qualtrics survey URL. Could I do it from the platform and with the brand admin account?

If you are just redirecting to the Qualtrics survey URL, you won't need the brand admin's account. Just pull the Anonymous link from with Qualtrics Distributions.

Thank you very much! All this is information was very helpful !