The best resource for this is Foundation Center http://www.fdncenter.org . They have a very good database covering all organizations giving grants and you can find foundations that are interested in supporting your type of cause. However, it’s a subscription based website ($19.95 per month I think is the cheapest subscription price). But it certainly is worth the fee. You can subscribe one month, and make sure you get all the info you need and then stop the subscription.

If you want to try federal sources, you can go to the Catalog of Federal Domestic Assistance (CFDA) http://www.cfda.gov and Grants.gov http://www.grants.gov – these are two sites created by the federal government to provide transparency and information on grants.

CFDA has several ways to search for grants. I suggest you search by beneficiary and select Private Non Profit Organization and you can find listings of grants available http://12.46.245.173/pls/portal30/CATALOG.BROWSE_BENEF_RPT.show

However, to be able to get grant awards, you need to be 501C(3) certified from the IRS http://www.irs.gov/charities/charitable/index.html
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If you need money to build a web site for a nonprofit organization, you need to talk to the organization about what those costs are going to be: a host for the web site, URL(s) registration, etc. If you want to be paid for your time, then you need to make a proposal to the organization, who may or may not have money to pay you. If the organization wants to seek funding for the organization, to then use to cover the costs of building this web site, that’s up to them.
References :

http://www.grants.gov/search/search.do;jsessionid=LNSWs6gZTqssXXgyJnKfX7Jmz1kKNJSMRykbMv7zw1MrnYM6451h!1636491997?mode=PAGECHANGE&pageNum=3

Check that for some more info
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