According to Bright Lights, Dark Shadows by Carl Magnus Palm, ABBA appeared on this show in 1974.

However, further research since publication of the book shows that it's extremely unlikely that they did appear on this show after all. ABBA arrived in the US on the evening (Swedish time because their plane left in the "morning", Swedish time) of Sunday 29 September 1974 according to reports. This would mean that it would be afternoon in New York when they actually arrived.

It's extremely unlikely that they would be booked to appear on a live television broadcast on the day of arrival and the plane probably hadn't landed in time for them to make it to the studio in time (if it was a LIVE broadcast).

Wonderama was broadcast that same day, presumably much earlier in the day (because it was a children's show). "Two films" were shown on the show, which in all likelihood would have been the promo clips for Waterloo and Ring Ring and therefore they probably didn't appear in person.

In June 2007 I was contacted by a US ABBA fan who said, "I can confirm without any hesitation that yes, ABBA did perform live on Wonderama! I was 13 in 1974 so I can't confirm the date of the appearance, or even the year, but I have just been thinking back to the first time I saw ABBA on Wonderama when I was a kid! Wonderama was only televised on Sunday mornings, and it was on for two or possibly three hours. It was a live show for children and teens that aired on New York's TV channel 5 which was NEW at the time. Two things stand out in my memory of the performances (I believe they lip-synched two songs). First was how beautiful the girls were in their bright sparkly outfits, and secondly I firmly remember the "star guitar" that Bjorn played during the performance. They lip-synched the two songs live in the studio in front of the kids in the studio, it wasn't just a showing of music videos. I would love to see some video footage of that performance again as it would bring back some great childhood memories!" He wrote further, "I did a little online investigation and found out that Wonderama wasn't live. It was a taped show. Here is a little bit of a posting from an assistant to the host Bob McAllister: The taping didn't start until 13:00. I don't know where the person who wrote in came up with 9:00, but we weren't even allowed to have parents or kids line-up until after Midday Live went off the air at noon. Also, the parents who didn't have to stay were kept in a spacious room on the third floor (Kluge Hall), with easy chairs and a couple of big monitors and soft drinks. We tried to make those who did stay comfortable. Very often, I would go down with other staff members, and allow those parents to come to the fifth floor and peek into the studio and/or control room. Against the rules, but nice to do."

Extra comment from Carl Magnus Palm: "I have a report from the time when it actually happened that clearly states that “two filmed segments” were shown. It may not have been the promo clips; perhaps they came from Star Parade as you suggest. All reports from this visit also seem to suggest that they weren’t really there to “perform” as such, but to do meet-and-greets, be interviewed and so on. And, also, according to newspaper reports, they didn’t arrive until the day of the broadcast, so if that is true I don’t see how they could have been on it."

Who can confirm one way or the other finally?!

Wonderama, 1974;
Wonderama (Mamma Mia, SOS and Fernando), 1976;

Thanks to Yuri, Carl Magnus Palm, Steve Layton and Warren Zulawski