I got my grantwriting experience through staff positions in community relations and fundraising. Are you trying to get contract work or a staff position as a grantwriter? That's hard without experience b/c organizations want to know what successful grants you've written.
If you work for or involved with a nonprofit, try that organization first. If you have children, don't forget to ask at your child's school. You might want to do some research and come up with a grantmaker whose criteria fit the organization. I find that it's the research more than the writing that nonprofits can't find time to do.
Some nonprofits would probably accept a volunteer grantwriter. The one I work for wouldn't unless the person had a good bit of experience b/c grantwriting is too important to the organization to risk someone inexperienced. If you try this route, go to small nonprofits with little staff b/c they will really need the help. You might also ask your United Way for suggestions on agencies that could use your help.
Good luck.