I don't know much about first families and can't seem to find an answer to this question. I know in other countries, family of current leaders have had combat positions, but has a President's immediate family ever served while being part of the First Family?
Yes. In World War II, FDR's sons served in the military, most of them in combat units.
James Roosevelt, the eldest son, had received a direct commission as a Lieutenant Colonel in the USMC in 1936, which was a matter of some political controversy at the time. After the European war broke out in 1939, he resigned this commission and was again commissioned as a Captain in the USMC Reserve. He went on active duty in 1940. In April 1941, he went on a secret tour to assure friendly governments the US would be entering the war, and during this time, came under attack by Axis forces. When the US formally entered the war, he was assigned to a Marine Raiders unit and became 2nd in command of Carlson's Raiders. Later in the war he served in a number of staff positions.
Elliott Roosevelt, the second oldest surviving son (one of FDR's sons had died as an infant) received a Captain's commission in the US Army Air Corps in 1940. He served in, and later commanded, a number of USAAC reconnaissance and aerial photography units.
FDR Jr. was commissioned as an ensign in the US Navy in 1940. He served aboard US destroyers escorting convoys in the Atlantic, and was eventually promoted to Lt. Commander and received command of the destroyer escort USS Ulvert M. Moore in 1944. His vessel was credited with destroying two Japanese aircraft and a Japanese submarine while under his command.
FDR's youngest son, John Aspinwall Roosevelt, was initially a conscientious objector. His family convinced him to join the US Navy and he was commissioned as an ensign in 1941. Initially he was assigned to shore duty in a supply capacity, but eventually did go to sea and served on the carrier USS Wasp during combat operations.
In World War I, Woodrow Wilson's children were all female and so none of them served in combat.
I also am a bit remiss because I missed one of the more interesting cases - John Eisenhower. He was the younger son of Dwight Eisenhower, though his older brother Doud Eisenhower had died of scarlet fever as a child. John attended the United States Military Academy and graduated on D-Day.
John's status as a regular army officer presented a conundrum as his father was at the time Supreme Commander of Allied Forces in Europe. His commanders were reluctant to put him in harm's way over concerns for his safety, which potentially harmed his career considering the expectation for career officers serving in World War II. This again became an issue in the Korean War, with his father becoming president at the end of the war. John initially served in an infantry battalion during the Korean War, but was reassigned to a safer position at divisional HQ. He went on to serve as a regular officer until 1963, and a reserve officer into the 1970s.
Perhaps the most interesting thing is that we have John Eisenhower's own opinion on the matter, from a NY Times article published in 2008. I hope this doesn't run afoul of the 20 year rule, but you can find the article here: http://www.nytimes.com/2008/09/28/opinion/28eisenhower.html
While he didn't actually serve in combat, I think Robert Todd Lincoln is an interesting example. He wanted to serve in the Union Army throughout the war, but Mary Todd refused to allow it until February 1865, just two months before Lincoln was assassinated. So, Robert Todd served in the armed forced for 2 months during the Civil War as a member of the First Family, but did not actually serve in combat.