You can start looking whenever you want. I am a nanny for a family where the dad is a high school teacher which means I have the summers off as he is home for school. I always start looking for summer nannying jobs sooner rather then later to make sure I can fill my time.

1st PP here. I wouldn't start looking until April/May if you want a June start

Start on the websites listed above, but I don't think you should start looking until a bit later in the school year. Most people don't know what their summer schedule is going to be like a year in advance. I found my first summer nannying gig in November of the year prior, and so many things had changed by the next summer that the position didn't end up working out. The MB was offered a WFH position, and I ended up taking summer classes. I think January should be the earliest that you start looking.

care.com
sittercity.com
craigslist

After enduring a crazy summer camp and swim team schedule, I think we should get a summer nanny next year. Where do I find one?