Whether coming from the west or east on I-70, take Exit #180 off of I-70, about 4 miles east of the main Vail town exit. Proceed to the north side of the interstate and turn east (right) onto Fall Line Drive. Drive just a short distance past some condo units on your left, cross Pitkin Creek and find a very small trailhead parking area immediately after crossing the creek. Just a few yards further is a cul-de-sac and another condo unit. Do not park there. The trailhead parking offers a total of 10 slots. Obviously, these will fill on weekends. If you can't find a parking spot, try going back to the Exit 180 interchange and drive south on Big Horn Road then SE and there's a parking lot just past the interchange you may be able to use. From there, you'll have to walk back to the trailhead.
The nearest campground is the Gore Creek Campground, a National Forest Service fee area with vault toilets and water. To get there from the Exit #180, you'll need to drive SE on Big Horn Road past the trailhead parking for the Deluge Lake and Gore Creek TH. Reservations may be made for campsites at reserveamerica.com. Other than that, there is virtually no other place close by with at-large/primitive camping. Your best bet may be to go up to the summit of Vail Pass and drive down the Black Lakes Road where there is a pullout parking area right where the road is barred from vehicular traffic and becomes the Vail Pass bike trail, east end. This is not a primitive camp location but you may be able to sleep in your vehicle.