With the advent of very affordable technology it would probably be a good idea to hire a college video production majoring student [or a couple] and pay them a small stipend. They would be motivated to do a good job to show to thier future clients and employers.
The biggest labor is in editing the video for production use. Probably a better idea to first offer archives of the games until you get some more money for "live" webcast and a dedicated team of two to three people to staff each game. I would probably just start offering only highlight clips initially. A video editor would have to go through about 3 hours of footage for each game to splice them out and compile them into something nice looking for the families as a keepsake. They love that stuff.
My own baseball league teammates absolutely loved seeing ourselves on video last year.
I used to be a league photographer for a men's very competitive softball league and it took me about 4 hours each night just to edit photos and post them on the website. I can imagine video editing is far more time consuming than what I did. My equipment consisted of a 4pixel fuji digital camera with a 8x optical zoom taking 1200x800 res shots.
If you choose to buy your own video camera equipment make sure it has at least a 32x OPTICAL zoom on it. Baseball fields are easily 50% bigger than softball fields.
Digital zooms are ghetto and shouldn't be used at all for quality work.
My not so humble opinions.