This is barely squeaking into late medieval and is probably early modern (pls u/sunagainstgold don't ban me) but the Chatham Chest is a good example of a scheme that's not quite insurance, but does involve paying into a fund for later reimbursement for disability. (It's a pension scheme.) Adapted from an earlier answer:

The English navy started a specific fund to care for injured and disabled seamen called the Chatham Chest. It was named after that for the, well, large iron-bound chest it was kept in, it was meant to offer lump sum and annual payments to disabled seamen. It was originally established around 1590, likely by Charles Howard (the first Earl of Nottingham, as well as Lord High Admiral of England during the Armada), to help sailors injured in the Spanish war.

The chest functioned as a pension scheme for disabled seamen. Every sailor in the Navy paid into it 6d per month (after 1649, surgeons and chaplains paid 8d and 10d), and it disbursed annual payments to wounded men based on their injuries -- rates varied over time, but they could be as low as £6 for loss of a limb or up to £15 for loss of two limbs, plus immediate money when the sailor sustained the injury (or, realistically, when he was injured, lived, and applied for relief).

The chest, being just a huge box full of money kept at the main naval yard, was sometimes stupendously mismanaged; we know for example that Sir Sackville Crowe, Treasurer of the Navy from 1627-29, "borrowed" £3,005 from the chest which was never repaid, even after he was sued for the money. (To be fair to Sir Crowe, during James' reign the navy was strapped for cash; many men had their pay in arrears for six years or more.) But, it was meant to function as a relief for the "common sailor" and did so, with varying degrees of success, until 1803 when it was merged into a similar pension scheme at the Greenwich naval hospital.