A few things that could be blocking this in your situation is having your messages to Friends Only or Nobody, they are blocked, or etc. However, I don't see how they would send you the club invite if you blocked them, but I don't know.
Unable to message?

Hello chess people! I am an active chess.com member, and I have many interactions with other players here on chess.com (and irl). I talk to friends on here, I accept club invites, and I give tips to those who ask. However, I have noticed that I cannot respond to a certain message asking me to join a club. I joined the club, and I was going to tell the person who invited me to the club that I joined, but I saw a message at the bottom of the page that said “You are currently unable to message with this account”. I have no evidence that the person blocked me, and I can still message my friends on here, so what happened? Do some people have a setting that blocks contact with other users unless they are friends?
If you know anything about this, please let me know! Thank you!
You're blocked by that member or they have their messages to Only Friends or Nobody

Hello chess people! I am an active chess.com member, and I have many interactions with other players here on chess.com (and irl). I talk to friends on here, I accept club invites, and I give tips to those who ask. However, I have noticed that I cannot respond to a certain message asking me to join a club. I joined the club, and I was going to tell the person who invited me to the club that I joined, but I saw a message at the bottom of the page that said “You are currently unable to message with this account”. I have no evidence that the person blocked me, and I can still message my friends on here, so what happened? Do some people have a setting that blocks contact with other users unless they are friends?
If you know anything about this, please let me know! Thank you!
You're blocked by that member or they have their messages to Only Friends or Nobody
Okay, I think it must be their settings. When I click on their profile, it doesn’t say that they blocked me. Anyways, thanks for the answers!

There are such players who block messages from “non-friends”. Probably they do it temporary to pass moment when somebody writes them constantly. And they do not want to communicate.

Hello chess people! I am an active chess.com member, and I have many interactions with other players here on chess.com (and irl). I talk to friends on here, I accept club invites, and I give tips to those who ask. However, I have noticed that I cannot respond to a certain message asking me to join a club. I joined the club, and I was going to tell the person who invited me to the club that I joined, but I saw a message at the bottom of the page that said “You are currently unable to message with this account”. I have no evidence that the person blocked me, and I can still message my friends on here, so what happened? Do some people have a setting that blocks contact with other users unless they are friends?
If you know anything about this, please let me know! Thank you!
You're blocked by that member or they have their messages to Only Friends or Nobody
Also happens when you are muted or just muted in messages
Hello chess people! I am an active chess.com member, and I have many interactions with other players here on chess.com (and irl). I talk to friends on here, I accept club invites, and I give tips to those who ask. However, I have noticed that I cannot respond to a certain message asking me to join a club. I joined the club, and I was going to tell the person who invited me to the club that I joined, but I saw a message at the bottom of the page that said “You are currently unable to message with this account”. I have no evidence that the person blocked me, and I can still message my friends on here, so what happened? Do some people have a setting that blocks contact with other users unless they are friends?
If you know anything about this, please let me know! Thank you!