I have cycled southern Utah and the Grand Canyon a half dozen times.
I never had reservations and never had problems except for Zion NP. (Mt Zion???)
I'm not sure from your rather limited post what you mean exactly.
It's like 200 miles between the North Rim and the South Rim on the highway.
You can hike across the canyon and ship your bike on the shuttle service.
(I've done that four times)
However, you will need to get a bunkhouse bed at Phantom Ranch -
Or a backcountry campsite at Bright Angel Creek.
These are EXTREMELY hard to come by.
Both the North Rim and South Rim have hiker/biker campsites.
The North Rim hiker/biker site is one of the nicest anywhere.
If you a biking from North Rim to South Rim it will take 3 days.
It's 80-plus downhill miles to Marble Canyon - lodge/RV Park.
Another 80 miles to Cameron - historic trading post/RV Park.
Then a big uphill to the South Rim.
US 89 between Marble Canyon and Cameron has lots of traffic -
Plus the shoulder disappears whenever they make a passing lane.
(Which sucks since cars are going even faster to pass)
Zion NP bites.
Not only do they NOT have hiker/biker camping,
But you also have to beg a ride through the tunnel.
(Do bikes make it unsafe for 50 foot RVs or vice versa?)
I have written Zion many times.
They have an $80 zillion natural gas bus system -
But it would be a lot cheaper to get people on bicycles.
Then there's the trek between Zion and the North Rim.
Kanab makes a good first stop.
The campground is funky and loud - there's also a nice hostel.
I have always biked it from Utah to Arizona BUT !!!
There is usually a moderate to strong SSW wind.
From Fredonia to Jacobs Lake and from Marble to Cameron -
Can be brutal - also tough from Jacobs Lake into North Rim.
If you are doing this between St. George and Flagstaff -
You might want to consider the opposite direction.