An employer can decide what equipment to provide for the workers. In most cases the employer provides the necessary equipment. There is nothing illegal in the employer saying to get or keep the job the employee must pay for a personal computer and internet access.
We have freedoms in this country so you may freely decide to work for someone else with better work requirements. You are an 'at will' employee so you have that absolute right in our system.
Good luck.
It is not illegal. Think of it like a car (or other prerequisite for a job); many employers require that their employees own a vehicle, but will not pay for the vehicle. If you do purchase a computer for work use, however, you may want to consider speaking with an accountant or tax attorney regarding any possible tax benefits to the expense.
This answer does not constitute legal advice, nor does not create an attorney-client relationship. I am licensed to practice law in Florida only and this answer should not be construed as an offer to practice law outside Florida.