Operation Flipper, in November 1941 was a British special forces operation aimed at killing or capturing Erwin Rommel.

The intention was to disrupt the German Command and Control infrastructure before the start of Operation Crusader, which was intended to relieve the siege of Tobruk. It was felt that Rommel was such a pivotal figure for the German army in North Africa that his death might sway the result.

The operation failed, because Rommel had left for Rome ahead of the attack (almost half the attacking force were unable to get ashore due to bad weather which meant that the other mission objectives were also not achieved).

Lt. Colonel Laycock's report on the raid, dated 5 January 1942, can be read at the UK National Archives (reference WO 201/720)

EDIT:

As a side-note in regard to Operation Vengeance, I found two papers that you might find interesting

The first of these is the 2015 monograph Killing a Peacock: A Case Study of the Targeted Killing of Admiral Isoroku Yamamoto, by Maj Adonis C. Arvanitakis, United States Air Force, submitted to the School of Advanced Military Studies at Fort Leavenworth, which looks at the planning and execution of the operation in some detail.

The second paper is the 1992 thesis Bullets With Names: The Deadly Dilemma by Roger G. Herbert, Jr., Lieutenant, United States Navy, which considers Operation Vengeance in the wider context of targeted assassinations from the perspective of the United States.

The operation to kill Bin Laden (2011)

Yes, in recent memory the US Millitary started an operation specifically to take out Osama Bin Laden (it happened in 2011):

"The Associated Press reported at the time two U.S. officials as stating the operation was "a kill-or-capture mission, since the U.S. doesn't kill unarmed people trying to surrender", but that "it was clear from the beginning that whoever was behind those walls had no intention of surrendering""

https://en.wikipedia.org/wiki/Death_of_Osama_bin_Laden#Operation_Neptune_Spear

Attempt to kill or capture Tito

Operation Rösselsprung was a failed attempt by the Germans to capture or kill Marshal Josip Broz Tito of Yugoslavia on the 25th of May, 1944.

The attempt to kill or capture Tito was led by Kurt Rybka with 500th SS Parachute Battalion. Tito, however, escaped from his cave headquarters after the German's first assault had failed. The only part of Tito they ended up capturing was his uniform...

"Tito with members of the Supreme Staff in front of the cave in Drvar , May 1944." This photo was presumably taken not long before the German assault. Source: Wikipedia

Initially involved in the attempt was Otto Skorzeny, perhaps best known for his key role in the rescue of Mussolini in Operation Eiche in September 1943. Skorzeny's plan, though, was compromised and had no further part in the events which followed.

Capture of Admiral Horthy, regent of Hungary

Perhaps a marginal case is that of the October 1944 Operation Panzerfaust, the successful capture of the Hungarian regent Admiral Miklos Horthy, which also involved Skorzeny.

In 1920, Horthy had originally accepted the regency on the condition that he was also commander of the armed forces, but I can find little evidence as to what extent he was involved land-based operations. He did, however, continue to wear a uniform.

The United States opened the 2003 invasion of Iraq with an attempted Decapitation Strike against Saddam Hussein.

I just learned that Operation Gaff (1944) was another such attempt of taking out the “usual” Erwin Rommel.

The abduction of German General Kreipe on Crete in 1944 (documented in the book and movie Ill Met by Moonlight might be one, although the intent was not to kill. The general was captured and smuggled off of Crete as a prisoner of war.

The Polish Home Army (a military force still subservient to the Polish Government in Exile), had operations that took out various police and military officials - and even attempted to kill Hitler, although the operation failed.

https://en.wikipedia.org/wiki/Home_Army#Assassinations_of_Nazi_leaders

If we count the Ukrainian Insurgent Army (1942—1949 span of activity) (they had between 20.000 and 200.000 fighters) as a regular army. They actually killed many soviet officials, either from the army or the NKVD. Their more important kill was Nikolai Vatutin, commander of the 1st Ukranian Front during WWII.