The best way to get a Chihuahua puppy for free is to adopt from a local animal shelter or breed rescue group. Some of these groups feature fee-waived adoption programs.
Even if there is a small adoption fee, you are still essentially getting the puppy for free. Reputable rescues and shelters do not adopt animals out unless they have been vaccinated, spayed or neutered and microchipped. Some also come with a voucher for a free initial veterinary visit or other perks. These are necessary procedures for the health and safety of the puppy, and paying a veterinarian to do them is significantly more expensive than most adoption fees.
To increase your chances of finding Chihuahua puppies for adoption at all-breed rescues and shelters, check in frequently. Use a pet adoption website, such as Petfinder.com, to help your search. Purebred puppies are often in high demand and may be adopted quickly, so if you see an available puppy, get to the shelter as soon as possible. Consider adopting an adult Chihuahua instead of a puppy, because they tend to be available more often and may be easier to care for and train.
Contact Chihuahua rescue groups, even if they do not have any dogs available that meet your criteria. Let them know what you are looking for and ask to be alerted if puppies become available.