From either direction on I-70, take exit 180 about 4 miles east of the main town exit for Vail. As you exit, go south and onto Bighorn Road. Drive east on that road .8 mile and turn left onto Columbine Drive. This road is easy to miss and climbs steeply uphill, goes back under the interstate and then comes to a very small trailhead parking area amid some condo units. There are only six parking slots. If unable to park here, you may need to go back to the Bighorn Road/Interstate interchange area where there's a parking lot on the south side of the interstate, just as Bighorn Road makes its turn to the SE.
A few miles east of the trailhead parking back on Bighorn Road is the Gore Creek Campground, a National Forest Service fee area with vault toilets and water. 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.