California Labor Code Sec. 2802 requires employers to reimburse employees for "all necessary expenditures or loss incurred by the employee in direct consequence of the discharge of his or her duties [...]".
In the District of Columbia, D.C. Municipal Regulations, Title 7, Sec. 7-910.1 requires employers to "pay the cost of purchasing and maintaining any tools required of the employee in the performance of the business of the employer."
Under Federal Labor Standards Act, an employer only needs to reimburse their employees for work-related expenses that drop their earnings below the minimum wage. If the work expenses put the employees' wages under the minimum wage threshold, this may violate federal laws.
So, the answer is Yes in California and D.C. If and to the extent that the internet is "necessary," "in direct consequence," and "required for" the employee's work. Is the internet necessary for employees' work? How about the telephone bill, utility bill, gas for the car, laptop or computer usage, office supplies, or the morning coffee? Some of these expenses may be necessary, but some are superfluous.
It is important to implement policies that specify what expenses are "necessary" and what are not. Though a written policy alone will not make expenses unnecessary, employers may avoid the risk of lawsuits by auditing their practices and updating their policies.