It's definitely o.k. to visit in summer and you'll find that you're not alone. Folks with children need to go when they're out of school.
As stated, the campground in the park will be closed all summer this year and there will be road construction. You'll want a park with electric to be comfortable and there are plenty available.
If your family won't mind this short time period of getting up early then it's easily doable. Early is a gorgeous time to walk the trails. If there's a full moon, the hike to Delicate Arch is perfect for the sunset... many folks do this. Depending on the children's ages, definitely sign up before getting there for the 'Fiery Furnace' ranger-led hike. It's very popular and you need to sign up in advance. It's a fun hike through the hoodoos and you'll have some shade. Read about it on the park's website.
https://www.nps.gov/arch/index.htmThere's a great flat bike trail starting in town and going down the river road. A boat trip on the Colorado River will be somewhat cooling. Naturally, 4-wheeling is popular and if you don't have a proper vehicle, perhaps renting one or taking a tour could be done. They are fun, beautiful trails.
Hope you can go. It's an awesome place.