I was handed over the management of an existing website EQscience.com.
The nameservers are pointed to Cloudflare, but my customer (who had the site built and managed by another service supplier before me) has no details whatsoever on this account and since the previous management company no longer exists, he can not request them there either.
How can I go about this, so that I can add the domain to my account?
I don’t expect that support will be able to go through account recovery and give you access to the account as you never owned it. I would think that you will have to abandon that account and move the domain to your own account.