It's becoming more and more common in bios and other places to see people specifying the pronouns you should use to refer to them:
- "they/them" (known as "singular they")
- "xe/xem" (and other neopronouns)
- "she/they" (meaning either the "she" set of pronouns or the "they" set of pronouns)
(The format sometimes varies but two pronouns separated by a slash is the most common way to see it written.)
In the context of this developing culture, where more and more people identify as something other than their birth sex, "no pronouns" is unambiguous. It means that the person doesn't want to be referred to with any of the pronouns above. The reason for that will vary from individual to individual, but shouldn't ultimately matter. Try to respect the request, as for some people it is very important.
Instead of using "he", "she", etc., use the person's name, nickname, or a noun phrase like "my colleague" (as appropriate).
If you mess up, that's understandable because of how English is. According to what I've heard, you shouldn't make such a big deal out of it: just correct yourself and continue. It's certainly not easy, but you can make it a little easier for yourself by practicing not using pronouns ahead of time.
To clarify, you will still address a person like this as "you" when in person, "we" if you're collaborating together, or even "they" when it's a group including that person and others. It's just that you should avoid using 3rd person singular pronouns (or singular they) for such a person. (It also indirectly means you shouldn't use gendered nouns either such as "woman", "son", and so on.)
See also What do you do when someone doesn’t use any pronouns?