The best approach to getting donations is to always focus on what's in it for the business. What they want is the exposure. Supporting a charity makes THEM look good in the eyes of potential customers. Be sure to mention your sponsors in your advertisements, and make sure your sponsors know they'll be mentioned.
The big thing they are getting from associating with you is free advertising and PR. The best donation ideas are ones where they can donate something that would make a big difference to you, but doesn't cost them much out of pocket. A few thoughts are:
1. Radio advertising. The radio station has to pay the rent and salaries anyway. It doesn't cost them much to run another ad. Plus, in many cases, they have to run a certain amount of "public service ads."
2. Restaurant coupons. Again, it doesn't cost the biz much out of pocket, but a free meal is a big draw for things like raffles or auctions.
3. Printing. Many times you can get your flyers or products (like a calendar) printed up as a donation, particularly if you know of another company that's printing something at the same time. You can do what they call a "gang run" where your print job goes through the press at the same time as another. It costs the printer almost nothing except paper.
4. Random products. What's "trash" to one business may be treasure to you. Inventory that "isn't moving" can be another source of great raffle items. It can be cheaper for the business to donate an item to you than to continue to store it.
As for how to go about it, you might start with a letter or email and then follow up by phone or in person. The best thing is face-to-face meetings. The larger the donation you are going after, the more personal "face time" you're probably going to have to put in.
Years ago, for a big fundraiser for an equine therapy organization, friends of mine walked down the main street of our tiny town and stopped into every single business. In one afternoon, they got an amazing amount of donations for a raffle and auction. It is possible, but it's work, and you have to deal with quite a lot of rejection too.
It's helpful to start with people you know personally. Try brainstorming with people in your organization and see who knows business owners. Make a list and start with people who know someone in your organization directly.