Yes! Not a big issue. Himachal as compared to Uttarakhand is much safer during monsoons. But it also depends where do you plan to go. Places like Spiti or higher altitudes get a little risky.
With family, it can get a bit tricky with the roads in particular, landslides, Uninterrupted rains and so on. So make sure you check the weather in advance and also make sure you don't start immediately after some very bad weather condition for that could mean disturbed and possibly caved-in roads that could be dangerous it delay your trip while enroute. Agreed that you can't really trust the weather forecasters, yet a simple comparison with last year's weather and the teens this year will give a fair idea. If there sky is clear or only shows scattered rains during the window of your journey, go for it for it is quite pleasant in July all over Himachal.