The definitive answer is no. Italian military leaders were not charged for their conduct during the Ethiopian War. That isn't to say that there wasn't any international backlash against Italy's invasion across the world stage, which mostly included sanctions and embargos of exports to Italy. After WWII however, leftover loyalists to Mussolini were tried for crimes committed in Ethiopia and WWII, but the sentences given to them leave much to be desired.

The orders to use chemical gas as a tactical weapon against the Ethiopians was authorized by Mussolini within 4 weeks of the start of the war, where orders were given to Field Marshal Rodolfo Graziani, commander of the southern front of the war as follows:

To His Excellency Graziani. The use of gas as an ultima ratio to overwhelm enemy resistance and in case of counter-attack is authorized. Mussolini.

2 months later, the same order was issued to the Northern Front, under Marshal Pietro Badoglio:

To His Excellency Badoglio. Given the enemy system I have authorized Your Excellency the use even on a vast scale of any gas and flamethrowers. Mussolini.

Ethiopian diplomats, and later Emperor Haile Selassie himself appealed to the League of Nations to end the conflict and bring justice for Italy's war crimes against the Ethiopian populace, pleading to international delegates to take action against the Italian invasion, but his hearing was detracted by the Italian journalists within the audience who heckled, jeered, and blew whistles during his speech to detract him. In the end, the League approved economic sanctions against Italy, but even these were ineffective as the sanctions did not prevent the export of military goods to Italy such as steel and oil, allowing Italy to continue building its military.

What Selassie's plea to the League of Nations did do however was diplomatically isolate Italy. Most of the world community came to recognize Italy's victory and occupation of Ethiopia save for 6 nations (Mexico, New Zealand, China, Spain, the US, and the Soviet Union), but Italy exhausted all goodwill it may have had from Great Britain and France especially, leaving Mussolini's regime in a vulnerable and diplomatically isolated position that wasn't addressed until 1938, when Italy began improving relations with Nazi Germany as a counterweight to Britain and France's opposition to Italian foreign policy.

Fast forward to the end of World War II and the establishment of the United Nations, Ethiopia again submitted charges for the Italian leaders who used mustard gas to be tried in the new United Nations War Crimes Commission, presenting all evidence in Ethiopia's possession to the crimes committed by Field Marshal Rodolfo Graziani in particular. The Commission came to the conclusion that there was a case to be made against Graziani and his chiefs of staff, but representatives of Great Britain in particular chose not to pursue charges against Graziani, with the United States offering an ambivalent response at best. Emperor Selassie made a direct appeal to Britain, the US, the Soviet Union, and France to reconsider, but was rejected.

In the end, an Italian military tribunal did charge Graziani and sentenced him to serve 19 years in prison for collaboration with Nazi Germany during the Italian Civil War in 1948, but he only served 4 months of his sentence and was released, where he spent the rest of his life as the Honorary President of the Neo-Fascist Italian Social Movement, regarded as a direct successor to Mussolini's National Fascist Party until he died in 1955. Pietro Badoglio, the Marshall of Italy's Northern Front, was seen as a guarantor of Anti-Communism in Italy by the Western Allies, and war crime charges were never pursued against him.

In terms of monetary reparations, that's a different story. In the treaty of Peace with Italy signed in 1947, reparations were paid in the amount of $25 million to Ethiopia, but this amount was mostly paid in the form of goods and services, rather than cold hard cash.

Sources:

Mussolini by Denis Mack Smith

Italian Fascist War Crimes in Ethiopia: A History of Their Discussion, from the League of Nations to the United Nations (1936–1949) by Richard Prankhurst