Is it incorrect grammar to say that someone "sent to say" something? As in "Joe sent to say that he would love to join you all for the pic nic". I know that I don't have the greatest grammar skills in the world, but I thought that that was a proper saying.
It's grammatically correct. You'll get some arguments from others as to whether or not it is semantically correct.
When I have encountered and used the phrase, it almost always has an implied verb object (which depending on the surrounding sentences might be confusing), with the basic implication being that the means by which Joe sent is irrelevant or already established by context.
Areas where I've heard it used:
Did you send an email to the Johnsons? Yes, I sent to say to that we were coming.
I think that the best answer lies in a compilation of insights from the comments of Edwin Ashworth, FumbleFingers, and tchrist.
It's easily understood, but I have to say OP's version sounds rather dated (if not actually archaic). Far more common is he sent word that..., but as this NGram shows, even that is essentially phrasing from an earlier age.
From Edwin Ashworth:
Seriously, 'wrote to say that', 'phoned to say that', 'texted to say that' are all fine and idiomatic. Using the wording 'sent to say that' is dated to archaic (but not incorrect), as FF says.
Joe sent whom to say what? You have to send something, and you are welcome to send someone something, too, like sending someone to say something. But send is not catenative like go or come, and I cannot fathom how the to-infinitive can possibly work as a direct object for send without some indirect object involved. Are you sure you aren’t just mishearing Joe said to say that he’d be late or some such thing?
These answers may not seem to be entirely in accord, but I think that they give effective voice to the three main points worth covering in a thorough response to the OP's question. First, the phrasing "sent to say" sounds dated and has become less frequently used in recent times. Second, some still-current expressions (such as "phoned to say" and "wrote to say") follow the same form as "sent to say" and yet don't pose any problems of coherence. Third, nevertheless, the expression "sent to say" suffers from an unsettling lack of definition about what precisely the sender has sent—and no comparable shortcoming arises in the case of "wrote to say" or "called to say."
Perhaps the underlying difference is that idiomatically we connect the phrase "wrote to say" and "phoned to say" with "wrote [to me] to say" and "phoned [me] to say," whereas with "sent to say" we have to translate the phrase as something like "sent me a message that said" or "sent word to me saying." The upshot is that although I have no doubt about the meaning that "sent to say" attempts to impart, and although the phrase is ultimately (as Edwin Ashworth says) "not incorrect," it does its job rather awkwardly and has no apparent advantages over a number of alternative expressions.