Bruce shared some wonderful advice, this especially.
How can you make it a win for the sponsor? You can list them on your competition website as a sponsor for starters. How many participants, entries, and volunteers are you forecasting to have? That is your sponsors potential audience. Can they put a flyer in the score sheet mailing back to the entrant after the competition? Maybe they can give them a 10 minute blurb about themselves before the competition lunch. Maybe you can give them some time in front of your club at a future meeting. Perhaps you have worked with the potential sponsor in the past. Maybe you can do something for them in the future. If they are a new brewer maybe you can provide feedback on a distributor that you've worked with before. Maybe you can hope to arrange for them to host a tasting at a liquor store where you know the manager. Be prepared to be creative.
I can't add much, but I'll add my personal experience.
In soliciting sponsors and donors for our competition, it is good to make personal relationships
with people if you really want to go big. For instance, if you want to have every drop of beer donated for the competition or the judging gifts entirely paid for, you better get out there to the local breweries and sell them on why your competition would be great advertising for them. I have been meeting and schmoozing with industry reps, brewers and brewery owners for months now in order to get the items donated that we would love to have (but can't afford) at our competition in late February.
A cold email will probably not go very far in terms of larger donations, but if all you're in the market for is t-shirts and shaker pints, it'll probably serve you just fine.