WASHINGTON—The Senate on Monday voted to move ahead with a bill that could effectively end tax-free online shopping in most states, overcoming last-minute lobbying against the measure by antitax conservatives, online businesses and other critics.

Lawmakers voted 74 to 20 to open debate on the measure, which would allow states to compel out-of-state merchants to collect sales tax for them. Supporters say the existing tax break robs states of revenue. Opponents, including marketers such as eBay Inc., have argued it would raise...