In the (early) medieval periode, how would a soldier on campaign store his assets (loot, money, possessions he may have brought with him) both in battle and on the march?

It depends a bit on the Soldier, the place and the loot.

Usually, loot was stored in the camp with the other posessions of the Soldier, but if it was small and highly valuable, the soldier might store it on his person.

At the Battle of Tours 732 the Franks supposedly raided the Umayyad camp, and Umayyad soldiers broke rank to rush to the camp to protect their own loot, turning the battle in favour of the Franks.

At Visby 1361, the Danes had missed to loot one of their own before they buried him in a mass grave - he was found with a pouch containing no less than 126 Danish silver coins as well as some other coins, probably looted during the capture of the island of Öland earlier, so it is evident that some soldiers preferred to carry valuable loot on their person, even in battle.

The camp of most armies up to the late 1700s, when formal supply services were established, often contained more people than the army. These were the wives, girlsfriends and children of the soldiers as well as others (mobile tavernkeepers, alcohol peddlers, thieves and ruffians and others who could profit from following a looting army around). These handled cooking, laundry, tent pitching and camp establishment, caring for the sick and wounded and at times looting as well. Servicing the soldiers with gambling, drinking and prostitution was also very common.

Peter Hagendorf, a German mercenary in the 30 years' war, first for the Imperial side, then the Swedish, wrote a diary where he tells of his wife in the camp, how his comrades gives him a small part of their loot when he is wounded and how his wife goes out to loot when he is too sick to do it.