I've never found definitive sources for the Papal Military, and I found it intriguing that many people have not covered the Papal military. I have heard the army was relatively large, but no absolute number.
Part 1
The army of the Papal States during this period was usually only comprised a small permanent force supported by a larger militia; it was only during the crises of the French Revolution and the later Italian Unification Wars that large permanent forces were raised. Regardless of the number of men and the time period, the Papal army was generally regarded as one the worst armies in Europe
Towards a Professional Army
Prior to the War of Spanish Succession, there was virtually no standing Papal Army. Beside the Swiss Guard company and the two companies of cavalry (the Cavaleggeri and the Cavalieri di Guardia di Nostro Signore better known as the Lance Spezzate or Broken Lances) that formed the Pope's immediate bodyguard there were the small Guard Regiment stationed in Rome along with a Cuirassier company and a number of garrison companies that were spread around the major cities and fortresses of the state, including a large garrison in the port of Civitavecchia that also served on the galleys of the Papal fleet. There was also a battalion of Corsicans that had been raised in 1603 to police the border with Naples, but had evolved into a general gendarmerie role especially within the city of Rome. This was all backed by a large provincial militia (the Milizie Urbane) of a theoretical strength of 80,000 men including 3,500 horsemen, along with the militia bombadieri who would man the cannon within fortresses. Most of these troops were not paid, but rather received special privileges for serving.
The Corsican battalion was noted as being particularly undisciplined and prone to riotous behaviour which caused a diplomatic incident with France in 1662 after a fight with the soldiers of the French ambassadors guard. Against a background of rising tensions of France caused by the hostility of Louis XIV and Cardinal Mazarin towards Pope Alexander VII, this affair led to the mobilisation of the Papal army after France had invaded the Papal territories within France and threatened to march on Rome. The Pope wished to mobilise 20,000 infantry and 2,000 cavalry however only 14,000 were raised, including 2,000 from the militia. These would have faced a French army of almost 40,000 men however the Pope backed down and signed the humiliating Treaty of Pisa.
The War of Spanish Succession
The Popes had wished to remain outside of the wars between the Catholic states of Europe, however the conflict between France and Spain on one side and Austria and Savoy (amongst others) during the War of Spanish Succession meant that this neutrality could not last. With the war raging in Northern Italy and the road to Bourbon Naples running through the Papal States, the borders of the Papal State were continually threatened. Initial requests for assistance from the Duke of Parma in 1701 and the appearance of Imperial troops on the border meant that a large army was soon raised - regiments of infantry, horse and dragoons were raised mostly sponsored by nobles with the Papal States, but there were also foreigners attracted by the generous wages on offer. The previous army of around 5,000 men expanded to over 23,500 by the end of 1707 including 27 infantry regiments (of which 2 were from the Papal territories of Avignon and the Comtat Venaissin in France, 1 of German Catholics and one of Flemish and Walloon volunteers), 3 dragoon regiments and 1 cuirassier regiment. An army had moved to the North to the recently acquired town of Comaccio to guard against Imperial incursions, however the town was stormed and Papal forces lost over 1,000 men and evacuated the town. The army encountered great difficulties - there was conflict between the officers commanding the army and the Cardinal Legates who governed the territories, desertion and ill-discipline was endemic and corruption was rife amongst the offices, including (and especially) the Cardinals.
Reform
Defeat in the war and another humiliating peace treaty meant that army was reduced down to 5,000 men - practically all infantry except for 70 Cuirassiers. The garrison of Rome comprised the Cuirassiers, 7 companies of the guards and Corsicans and 220 men of the Castle St. Angelo garrison. There was also 300 men in Civitavecchia and 36 other companies scattered around the States. The previous regiments were dismissed or allowed to whither down and converted into companies. Reforms were attempted to improve the quality of the army, such as more training for bombadieri and a reduction in the size of companies to improve the quality of officers however these seemingly had little effect, however these attempts were stymied by continual cost-cutting within the Roman bureaucracy. The army was seen as superfluous and Papal power would be better exercised through diplomacy; this resulted in significant spending cuts to the army - spending in 1734 was only 60% of that in 1709 and by 1740 it was down to 45%. Army pay was heavily affected, particularly the men of the bombadieri and the Corsicans and the number of officers was reduced with a commensurate effect on army quality. Benedict XIV attempted to reform the army in 1740, partially by using household funds, however even these reforms did little to improve the performance of the army.