Franklin Duran of the Bolivian Drivers' Federation claimed that the strike was being adhered to by 90 percent of drivers, except in the eastern province of Santa Cruz.
He threatened to step the strike up a notch unless the government drops the proposed law, which Duran said could expose unwitting bus and taxi drivers to loss of their vehicles if drug traffickers used them to get around.
In La Paz, taxis were operating normally but bus service all but ground to a halt.
Interior Minister Carlos Romero asked for talks to head off the strike but it went ahead anyway, Duran said.
The lower house of Congress has given the bill preliminary approval. It still needs to go before the Senate and then be signed by President Evo Morales.
South America's poorest nation, Bolivia is also the world's second-largest producer of cocaine, after Peru, according to US figures.