Does using UpdraftPlus cause any cookies to be set for my website visitors?

No.

N.B. If you use the ‘Lock Admin’ facility in UpdraftPlus Premium to restrict access of admins to your UpdraftPlus settings, then successfully entering the lock password will set a cookie containing a cryptographic hash, for the length of time that was chosen in the ‘Lock admin’ settings. It is this cookie that enables the user to keep on accessing the page during that time interval, before they have to enter their credentials again. But since this has relevance only to back-end website administrators, it does not need including or evaluating for the purposes of your front-end cookie legal requirements.