The Order of St Lazarus was one of the smallest of the four international military orders and so made only a minor contribution to the crusades in the Holy Land and Mediterranean. It does not appear to have joined in crusade campaigns in Iberia or the Baltic.

The Order originated in the early twelfth century as a hospital in Jerusalem and by the early-thirteenth century it had militarised, fighting in the Levant alongside the Templars, Hospitallers, Teutonic Knights, and St Thomas Knights. Like these orders the brethren included knight-monks sworn to celibacy, poverty, and chastity who fought in the defence of the crusader states. But there was one significant difference about this order: many of the brethren were afflicted with leprosy. The hospital that the Lazarites originated as was a leper hospital and until 1253 it was custom for the masters of the Order to be lepers. In battle the Lazarites even fielded leprous knights. Around 1200 the legal code of the Kingdom of Jerusalem required leprous knights to joint the Lazarites, as did the Templars for their brethren from 1260.

The Order appears to have had a military role by 1234, when a papal appeal to clear the brethren's debts described these as being accrued in 'defence of the Holy Land'. In the mid-thirteenth century the English chronicler Matthew Paris drew a map of Acre showing the 'military house' of the brethren (it's the house and church on the left of the map, within the outer wall and next to the sea). With the exception of the siege of Acre in 1291, when twenty-five Lazarite knights fought in the defence of the city, we do not know how many brethren the Order could field.

Unsurprisingly, knights afflicted with serious diseases make for poor fighters. Their first recorded military engagement was the battle of La Forbie in 1244, a major defeat for the crusaders. All the leprous knights were killed. The Lazarites were also present at Marsuna in 1250, when Louis IX of France was captured. In 1252 the Order had joined a royal army en route to Jaffa when it suffered another mishap. The master of the Lazarites had scouted some cattle and other loot in a nearby town and left the column without permission from the king. But after finishing his looting he was attacked by Saracens. Only four men escaped.

This was particularly disastrous for the Order. The following year, Pope Innocent IV was petitioned by the brethren, asking that they be allowed to appoint healthy brethren to lead the Order instead of lepers as 'all the leper knights of the said house have been miserably killed by the enemies of the faith'. At the siege of Acre in 1291 the Lazarites joined the defence of the city, alongside the other four orders present in the Holy Land. All twenty-five of the Order's knights were killed when the city fell in May that year.

The loss of the mainland marked the end of the Lazarites' military role for some time. After a possible stop-over in Cyprus (the evidence for this is slight), the Order withdrew to its European estates which had funded their activities in the East. Boigny in France became the Lazarites headquarters and crusading activity appears to have ceased. The English, Apulian, and Hungarian branches of the Order began to break away in the fourteenth century. In 1489 Pope Innocent VIII ordered the suppression of the Lazarites and for their properties to be taken over by the Hospitallers, then waging a naval war against the Ottomans from the island state of Rhodes.

However, Innocent's orders were not followed everywhere. In France and Italy the Order soldiered on. In 1564 the Pope assigned the Italian Lazarites the duty of fighting pirates and enemies of the Church. In 1572 the brethren were united with the new Order of Saint Maurice to create the Order of SS Maurice and Lazarus, controlled by the duke of Savoy Emmanuel Philibert. The brethren were mandated by the papacy to provide two galleys for defence against heretics, infidels, and pirates. Two galleys were based at Nice in 1573. In 1578 the French Lazarites wrote to the French king offering to reform their order and take on a military role again. In 1609, the French brethren were united with a new military order, forming the Order of Our Lady of Mont Carmel and of St Lazarus, which fielded two galleys in France's war with England in 1666. It also maintained a small fleet in the Channel to combat English piracy.

Beyond this period the record gets particularly scanty. The brethren continued to serve in France as a special faction of the French aristocracy and officer corps until the Revolution. The Order of St Lazarus still exists today, though in several different organisations, including self-styled ones. Their military contribution was never particularly large or always successful, but alongside the Hospitallers and Teutonic Knights, they are one of only three military orders to maintain both their military and religious roles well into the modern period.

Sources:

Malcolm Barber, 'The Order of Saint Lazarus and the Crusades', The Catholic Historical Review, Vol. 80, No. 3 (Jul., 1994), pp. 439-56.

Bullarium, Privilegiorum ac Diplomaticum Romanorum Pontificum, ed. H. Mainardi (Rome, 1733-62), vol. 4:3.

Rafaël Hyacinthe, 'Crisis? What crisis? The “Waning” of the Order of St Lazarus after the Crusades', in Helen Nicholson (ed.), On the Margins of Crusading: The Military Orders, the Papacy and the Christian World (Farnham, 2011), pp. 178-87.

David Marcombe, Leper Knights: The Order of St Lazarus of Jerusalem in England, c.1150-1544 (Woodbridge, 2003).

You may be interested in episode 96 of the AskHistorians Podcast from last month. /u/Rhodis discussed the Order of St. Lazarus, among others, though I can't recall if he answered your specific questions.