I don't have kids, but when I was doing customer service at home, the two companies that I worked for had schedules that you could sign up for that were convenient for you. So it really depends on who you sign on with. There were also sometimes issues with getting the slots that worked for me if I wasn't online right when they opened up the schedule.
I have worked customer service with a small child in the past....and then years later with 2 small children. It can be done (especially on clients with very few calls) but HIGHLY stressful. When I only had one child, I worked overnight only and it wasn't bad unless she woke in the middle of the night crying or sick. With two, it was a lot more stressful due to the background noise. Even in a room down the hall while using a noise-cancelling headset I would stress out because they would run right near the door screaming and playing. If you have a larger home with someone to keep an eye on your toddler, then it can be done. Good luck!