While it might seem like you're being forceful, this tactic helps both parties. If you see a photo frame that will work perfectly at your next fundraising event but you've already spent that money on catering, ask a member of your local chamber of commerce to help cover the cost of the frames. How do they benefit? With optional personalization, you could add your not-for-profit's name AND the company's name as a contributing sponsor. Less money from your budget, and an easy advertising opportunity for them.
Another option is to ask businesses to donate other goods, thereby freeing up funds to use toward the event or other charity aspects. Have a board member who owns the local grocery store? Ask for food donations instead of paying for catering for your event. You would be amazed at the connections some of your supporters might have and how they can benefit your organization.