In France, we have the SPA (Société protectrice des animaux), a national-wide association, recognised by the government as of public interest. The SPA maintains a network of dispensaries (dispensaires), where people with financial difficulties can take appointments for their pets. However, there are only 11 of these structures, unevenly spread on the territory (see the official map).
The veterinary schools (Écoles vétérinaires), which are grandes écoles, also offer medical care with more affordable prices. There are four schools, located in Toulouse, Nantes, Lyon and Maison-Alfort.
Many local or regional animal care associations are located in the whole country, and might also help.
Feel free to complete this answer, as I may have missed legal aspects, as well as important associations. Also, I did not provide anything for the overseas territories (DOM/TOM).