By controlling a whole country?
The Whitecloaks essentially ruled the kingdom of Amadicia, Ailron being nominally the king but de facto a puppet of the Whitecloaks who held all power in the realm. As such, they'd presumably have access to all of the country's wealth, whether from taxes, trade, or whatever else. If you run a whole kingdom, you're unlikely to be strapped for cash.
From their supporters, believers, and members?
The Children of the Light are essentially a religious organisation, and as such they must have some actual supporters as well as people who are simply too weak or scared to resist them. Some of their members are personally wealthy, like Galad Damodred from a noble/royal family, and they might consider donating to the cause they believe in enough to join up. Other rich or noble figures might support the Whitecloaks too, either because they believe in the cause or because they see some political advantage in it for themselves.
Like the White Tower, the Children of the Light have been around for a long time and accumulated a lot of power. That's somewhat self-sustaining: simply being an institution with a lot of longevity and power makes people more willing to invest in you.
We know the Whitecloaks aren't averse to using threats to get what they want. They can accuse people of being Darkfriends at the drop of a hat, even relatively powerful people such as nobles. They have a big enough army that even indirect threats from them would be taken seriously. I don't have any evidence that this actually happened, but it's certainly plausible that they could extort "donations" from rich people on the understanding that they wouldn't then accuse those people of being Darkfriends.
(The same question has been asked on Reddit before. Some of my thoughts above are inspired from that thread, as well as Mithoron's comments here.)