You can do it by using an HR solution like TriNet. We pay about $155 per employee, but it's worth it. It allows us to focus on our products instead of payroll, health insurance, compliance, etc. When you take all those into consideration and how much time it will take for you to do that yourself, the cost is insignificant.
An added benefit is that you can use your credit card as a buffer -- and stack up points!
Related answer from another question.
Another benefit of a complete solution like trinet is that your team can use WageWorks' commuter benefits to pay transit and parking expenses with pre-tax dollars. Depending on how each employee commutes, that's usually about $900 a year in tax savings for each employee.
Edit: Looks like I spoke too soon. As I was researching, I did come across http://www.payrollbycreditcard.com/ which required AMEX, but it did not look legit. But TriNet is clearly legit, and the answer Nishank links to also mentions using American Express with them. Maybe they operate differently than Visa / Mastercard, in a way that lets payment processors not charge as hefty a fee? In any case, upvote Nishank's answer, not mine.
Original: The problem is someone's always taking a cut. Stripe, PayPal, SquareUp... there's going to be some kind of payment processor in the middle, and I'd guess that the ~3% fee is going to lose you (or the payment recipient, which you'd have to pay back with a check or overpay to compensate in this scenario) more than you'd gain in reward points.