I've only played on one beer league team. After having one name for the first season (only three returning players from the season before), we had a talk about spornsorship. Our last game of our first season, we had an outing a a local bar (with solid food choices as well) and noticed they sponsored random sports teams. A couple conversations later and we had uniforms paid for an a promise to re-evaluate their investment a season later.
We would go to the bar with the few fans we had for beers or food after games for postgame talk and easily surpassed the initial investment they made in us. We now have the bar asking how much our league fees are and where to write a check to as well as having shirts made. Chances are, we may be playing for close to free this season and future seasons in exchange for their patch on our bright green (their color) jersey, having our team named after them, and to bring fans and ourselves to the bar after games.
Other options entail looking for local breweries or whatever to add a patch or name to the bottom of the back of the jersey.
Plan of action would be to look for bars in the area that sponsor sports teams that your team wouldnt mind spending time in after games.
Find the player on your team that is the most charismatic and best with his or her words. There will be some negotiating and smooth talking. For the bar, it is all business. Provide a business case to provide any funds to you. It doesn't have to be a formal presentation but having numbers and ideas and thought will go a long way. Like I said early, a bar already sponsoring a team is easier to persuade, but in the end for the sponsor, it's all cost-benefit analysis.
In the end, sponsorships do not matter unless you have a team you enjoy and want to grow with. I'm lucky enough to have stumbled upon a solid team early in my beer league career (24 yo; first season) with leadership (40 yo beer league vet from Philly did the talking and provided a foundation for how tenured teams run) to see where to go for it.