How do people generally input these custom operators/characters such as the vector method call and the set operator used in the article?

You usually don't need to. Perl6 supports both Unicode and ASCII (a.k.a. "Texas") versions of these operators: https://docs.perl6.org/language/unicode_ascii

To answer the question, though: in my case I use X11's support for compose key sequences (with Caps Lock as my compose key) whenever possible (TODO: figure out a reliable way to add new sequences to fill in the gaps; I've been meaning to be able to type Compose + S + H + R + U + G to be able to instantly pop in that one shrug emote without having to search for it all the time).

Emacs (and other editors presumably) could probably be programmed to replace ASCII operators with Unicode operators on the fly, though I don't know of any packages which do this.

You can do that with an .XCompose file in your home directory. See https://github.com/kragen/xcompose for more info.