Looking for a friend.
They hire an animal wrangler that provides the service of bringing all the rats, getting the rats into place on set and making sure all the rats get back home safely.Here’s a list of wranglers for hire.
You’ll also want someone on set from the American Humane Association so you can be sure your staff and the wrangler are in compliance.
Okay, good, that’s a start.
Where do the wranglers typically get their rats? What breeds do they use?
When Mommy Rattus norvegicus and Daddy Rattus norvegicus love each other very much…
The problem, as you might guess, is not how you get more Brown Rats but how you don’t.
Nm
I would guess in addition to the critter wranglers-for-hire, there is a considerable amount of CGI employed these days to multiply the numbers of said critters, thus enhancing the audience enjoyment of them.
I remember in Bram Stoker’s Dracula there was an entry for silverfish Wrangler. I actually had to look up silverfish to figure out what they were wrangling.
*The Silence of the Lambs *had an credit for Assistant Moth Wrangler and Stylist.
I doubt this. Actual rats would be cheaper than CGI, wouldn’t they?
Maybe (though CGI software is becoming pretty cheap now). But CGI rats go where the director says, and do what the director orders. It’s much harder to do that with real live rats.
:dubious: Gonna assume you didn’t mean you are seeking to meet new friends in a rat swarm, but that is kind of how your OP comes across.
In Werner Herzog’s remake of Nosferatu, he wanted to use gray rats but could only get pet white rats, so he painted them gray. Most of them licked themselves clean fairly quickly.
They don’t see you as I do
I wish they would try to’
I’m sure they’d think again if they had a friend like Ben
Even before CGI, you don’t need a lot of live rats to make it look like a swarm. Camera angles, stuffed rats, matte panting, and selective editing and you can make 20 rats look like 200.
Ah yes, I’m asking for a friend. 
A friend that can’t be directly attacked for asking a question because of automatic assumptions that some inhumane, awful and/or illegal fate will fall upon some poor little critters with this knowledge. I am reasonably sure that no animals will be harmed as a result of answers to this questions.
I went and looked at the IMDB listing for Willard (the original version), and saw this list. But I’m sure they’re out of the biz by now.
Moe Di Sesso … trainer: “Socrates” and “Ben”
Marc Mercury … animal assistant
Nora Di Sesso … trainer: “Socrates” and “Ben” (uncredited)
If you just need one or two ‘hero’ rats, like if you just need a shot of a single rat biting someone’s face or whatever, then you get an animal wrangler and a trained animal.
If you need more than 5 of them, it’s easier and cheaper to tie it into the overall hiring brief for the CG people to handle along with the other special effects. A program like Massive is designed specifically for dealing with semi-fluid mass-creature mechanics and is responsible for a lot of crowd scenes, mass animal movement (think a thundering herd of bison) or huge background war scenes. It’s trivial easy to create and then render out a ‘realistic’ swarm of rats or vermin these days with no real animals ever making it near the set.
In “Brokeback Mountain” they used effects to make 200 sheep look like 2000 sheep. They’d film the sheep in on place, and then in another place, and then in another place. Then they’d composite the shots together. Then add a bunch of CGI background sheep.
So for your rat swarm you have several shots of real rats for your hero shots, mix together, then add CGI background rats.