No, not bicycles
To put your foot down is to be unyielding and insistent. Phrases.com offers no citations, but says
This expression is thought to have originated in 16th century but it actually became popular and common in 18th century. It is observable that when you clump or tramp your foot on ground, it means to demand someone’s attention.
A commenter on actuarialoutpost wonders if “the origin started around the time the bicycle was invented. In order to stop, you had to put your foot down.” But wikipedia says “bicycles were introduced in the late 19th century” which is incompatible with a 16th century origin. Though it is an inventive hypothesis.