Did regular taxes cover it or would the dominant political figure/family have to foot the bill? Or was it more of a group thing where the merchants and nobles all chipped in when the need arose?
Formally speaking, the city-state governments did, through various forms of taxation. To simplify, there were two broad categories: permanent or regular taxes and extraordinary or special taxes.
In cases of extreme emergency, leading families could and did use their own funds to hastily hire an army, with the expectation that they would be reimbursed in the future. This was the time that leading families became bankers, a famous example being the families of Genoa: Doria, Spinola, and Taxis. The Doria family owned a large navy they hired to the Spanish, the Spinola and Taxis had holdings all the way up Milan, Palatine, and Low Countries.
I know that in the Low Countries and Spain, by 1400s-1600s, guilds were quite well established and they could argue their interests in state governments, along with funding. I believe this was also the case in Italy.