The tradition comes from the Edo period and it is closely related to an old custom of bringing a gift to a newborn baby boy at the first New Year celebration. That gift was a set of bow (hamayumi) and arrow. Today, the bow is seldom found (although I bought one from Asakusa Shrine), but the hamaya arrow can be found at any shrine...
Regarding the votive plaque, usually it is an ema with the animal representing the zodiac sign of the year - the photo is from 2011, so as you can see, most of the plaques are with rabbits. But that's not a general rule: if you look closer, you will also find... Hello Kitty!
EXIF info:
Nikon D300
Lens: 18-70mm F/3.5-4.5G
Focal Length: 63mm
Aperture: F/6.3
Shutter Speed: 1/80s
ISO Sensitivity: ISO 200