When looking to be hired for a work at home position, how you present yourself will have a lot to do with whether or not you get hired. Your main reason for wanting home employment may have to do with the fact that you are tired of trying to juggle a job and children. This is an important reason to you but a prospective boss is not likely to be interested in your personal hardships.
Rather, work the application in a different light. Speak in terms of how much more productive you can be when you are not concerned with family issues and sick children while at work. Allowing a prospective employer to know you are able to give more time when you work from home is what will draw his attention. Follow employment instructions to the letter and always act in a professional manner. If you find an online vacancy you are interested in, reply with your resume and cover letter.
Online jobs are abundant. There is one waiting for you. Just apply and wait for the contacts to come in.