Community Organizations
If you live in New York City, Materials for the Arts is a dream come true. They fill a 35,000-square foot warehouse with supplies that they have gathered and then invite local artists to come shopping for free. They even offer classes to show artists how to use some of their random odds and ends. See if your community offers any sort of support for artists. Even if you can't find a dedicated non-profit, artists can be generous with unused supplies and often give them away on Craigslist and Facebook groups.
Grants
Art professionals can apply for grants and funding to put on an art exhibition, conduct research, or simply help pay for art supplies. Here are three sites where you can search for opportunities. Search through the listings at no cost. They cover all the need-to-know details for applying--including entry deadline, fees, location eligibility, and more:
The New York Foundation for the Arts
Corporate Sponsorships
Approach corporations that have a link to your work. Companies may prefer artists in their local area, or who paint a particular subject matter. Businesses can get overwhelmed with applications, so check their website for any posted guidelines for the donation process. In general, it is easier to try getting goods and services for free or reduced prices than cash from businesses.
Here is an YouTube video of an artist showing some of the supplies he has received for free through corporate sponsorships.