Most times I've done research for posts about words that originated in comic strips, I've been able to find fairly consistent information about where and when they first appeared. The information I was able to find may not have been very plentiful, but at least it told a consistent story. Not so with "worry wart." It seems pretty clear that the phrase originated in a comic strip, but no one seems to agree on when. I think I've found the correct answer, though.
What all the sources I found do agree on is that there was a character called "The Worry Wart" in the comic strip Out Our Way, created by J.R. Williams. Out Our Way was what you might call a nostalgia comic. It centered around people who appear to live in the late 19th century in rural surroundings. Most of the characters are unnamed, though many of them reappear from time to time. The jokes generally revolve around relatable situations that the reader would probably remember from their childhood, or possibly remember hearing about from their grandparents when they were young adults in the old days. The strip started in 1922 and had a fairly long run, ending in 1957.
All those sources also agree that the way the term was used in the strip wasn't exactly the same way we usually think of it. Normally when we use the phrase, we mean someone who worries incessantly, possibly to the point of growing warts. In the comic strip, The Worry Wart was a small boy who, due to his actions, caused endless amounts of worry to his family, and especially his mother. While the meaning wasn't the same, it's still fair to say that the comic strip is the origin of the phrase, as the term had never been used to refer to a person before, if it had ever been used at all. Etymologists are free to quibble, but it seems pretty clear cut to me.
The issue of when the term first appeared seems to be a bit thornier, and there is a lot of misinformation about it on the Internet, even from otherwise reputable sources. Now, if we all agree that the origin of the phrase is the comic strip, then it stands to reason that all we would need to do is find which strip the term first appeared in, and we have our answer. This does pose a problem for some, however, as they may not have access to an archive of the strip, and they may not feel inclined to look through every single one to find it. After all, it was a daily comic strip that ran for 35 years. That's a lot of strips. This means that most places you look are most likely relying on information from other sources that they trust to have the right information. Unfortunately, most of them do not.
Many places put the first appearance of the phrase in 1956, most likely owing to the fact that this is what the Oxford English Dictionary says. This is odd, given that Out Our Way only ended a year after that and had been running for quite a while by that point. One would think that they must be using a different source for the origin of the phrase, but they don't. They still credit Out Our Way, and some places even call it "a 50's comic strip" (which is also odd, as by that point its popularity was waning). The mistake seems to come from someone confusing the publication date of the comic strip with the publication date of a reprint comic book. Dell Comics did publish a reprint comic in 1956 called "Out Our Way With The Worry Wart," which was part of a series of comic books that reprinted various newspaper strips. I could only find one Out Our Way issue, and I doubt a popular phrase would have spawned from a single issue of a comic book. Incidentally, there is another comic book with a character by that name who appeared over a decade earlier. All-Flash, issue 15, from 1944, included a character named Ebenezer Jones. Jones suffered from crippling anxiety, which was discovered to be caused by some kind of virus. An evil scientist decided to make it contagious in order to infect others with the anxiety (that's Golden Age superhero comics for you), earning Jones the nickname "Worry Wart." Even if Out Our Way wasn't the origin of the phrase, that issue of All-Flash confirms that the term was in use long before 1956.
Other places I searched are much more reasonable. Most of them are fairly vague, stating that the term first appeared sometime in the 1930s. Merriam-Webster is more specific, putting the first appearance in 1936, though it does contain the caveat that this was the first time it was used as it's currently defined in said dictionary, not necessarily how it was used when the term first appeared. That's fair, though as I said, it's unlikely that it would be used at all if it hadn't been popularized as a phrase earlier than that.
Finally, we have sources that appear to actually correctly date the first appearance of the phrase, which I will link below. I know that these are correct, because although others may not have enough patience or access to resources to find this out, I certainly do. I searched through every Out Our Way strip, beginning in 1936, and worked backward. The earliest one I could find is from March 6, 1929 (pictured above). There are strips from earlier years that feature a similar character, but he's never called The Worry Wart in those strips. To be honest, because of that fact, I stopped searching in around the middle of 1927. It's possible that the term was used earlier, but unlikely.
For more (good) information take a look here: