Should I make reservations or wing it?

Very aggressive schedule. I would recommend dropping the west coast idea and make your destination Glacier NP and the Grand Tetons. That would be five 10 hour days to get there and 5 to get back, leaving 10 days for exploring and a couple for just in case. You won't be seeing anything on the way there and back besides rest areas and gas stations and rolling into a campground late and leaving early won't make you any fiends. Use freecampsites.net to find a place to pull over every 500 miles. It'll get easier the further west you get. I think you'll find that after 5 hours of driving, it won't seem like a vacation.

For a comparison, we set out on the same trip last summer. From Cape Cod out to Ohio, then up to the UP of Michigan and out US2 to Glacier and Puget Sound. Of course being retired, we set a leisurely pace of 4-5 hours on the road then staying somewhere for 2 or 3 days. We set aside 11 weeks out and back and didn't make it to Glacier because of fires. We didn't have reservations anywhere when we started, but instead made reservations about a week in advance. The only time we didn't get into where we wanted was Labor Day Weekend.

I'm a big fan of winging it, but as other have mentioned, if you're going to popular places over a summer weekend, you should book it now.