While we're a lot more limited right now due to travel ball, we are big fans of National Park road trips. I'm not sure about your child's injuries, but we managed a couple of days in Rocky Mountain National Park last summer while our younger one was in a boot for a stress fracture. So, no major hiking during that visit, but there's enough to see off the main roads and more tame trails to make it fun. Also forgot to note that we visited Capulin Volcano NM on our way to Colorado, since finding places along the way helps. We have three ways we've approached parks trips. Just be mindful of parks that are requiring advanced reservations... Acadia, Arches, Glacier, Haleakalā, Rocky Mountain, Shenandoah, Yosemite, and Zion. I believe Muir Woods NM is also requiring reservations. The caves all seem to require some form of extra ticket as well. I know I had to do this for Carlsbad, I'm seeing it in my planning for Wind Cave NP and Jewel Cave NM, and I thought I saw someone saying the name for Mammoth Cave NP. This stuff is all pretty easy to purchase on the Recreation.gov app if timed well. You'll also want an American the Beautiful NP pass if you go this route for travel. So, here's our three methods of approach-
1. Pick a park or one specific area and focus on that. Example: Yellowstone/Grand Teton- fly into Jackson Hole, Cody, etc. - rent a car - and just focus on those two parks/surrounding areas. At this stage of the game, while there may still be some lodging left at Canyon in Yellowstone, your best bet is probably staying outside of the park in West Yellowstone or Gardiner (north entrance should be open).
2. Fly to your parks road trip - we've done a couple of these where we pick a starting/ending point, fly in, rent a car, and make a giant loop to hit an array of parks and monuments. Example: Fly into Denver, acclimate to the elevation for a day or two (explore foothills around Golden), drive to Estes Park for Rocky Mountain NP, drive to Rangely, CO or Vernal, UT to visit Dinosaur NM, drive to Moab, UT to visit Arches NP and Canyonlands NP, drive to Cortez, CO to visit Mesa Verde NP and Canyon of the Ancients NM, drive to Montrose, CO to visit Black Canyon of the Gunnison NP, drive to Alamosa, CO to visit Great Sand Dunes NP, drive to Buena Vista, CO to whitewater raft Browns Canyon NM, then back to Denver to visit the Rocky Mountain Arsenal National Wildlife Refuge. We did a couple of other like this that involved Badlands NP, Mount Rushmore NM, Little Bighorn NM, Crazy Horse, Devils Tower NM, Yellowstone NP, and Grand Teton NP. Another had us centered in Las Vegas and we went to Death Valley NP and Grand Canyon NP.
3. Drive from home- This will largely depend on how far you're willing to drive from home, since everyone has a different starting point and personal threshold for time on the road. We did a shorter one where we visited Hot Springs NP, Graceland, and Great Smokey Mountains NP, and then looped around through Atlanta and New Orleans before heading home. The biggest one we've done was about 6,500 miles and took us to the Oklahoma City National Memorial, Pipestone NM, Theodore Roosevelt NP, Glacier NP, Yellowstone NP, Shoshone Falls ID, Capitol Reef NP, Bryce Canyon NP, Grand Staircase Escalante NM, Zion NP, Glen Canyon NRA, Horseshoe Bend, Monument Valley, Mexican Hat, Shiprock, Petroglyph NM, White Sands NP, Carlsbad Caverns NP, and Guadalupe Mountains NP. I have a few others like this in my notes (one that takes us to Olympic and Yosemite...another that takes us to Big Bend, Petrified Forest and Saguaro) that we hope to someday work into our schedules.