I do not think so, as it was not designed for such. Going into orbit is one thing, bringing the pilots safely home is another. Consider this though. During the X-15 program, 13 flights (by eight pilots) met the USAF's criteria for a spaceflight by passing an altitude of 50 miles (80 km) and the pilots were accordingly awarded astronaut status by the USAF. NASA has also awarded astronaut wings to several X-15 pilots. Some respected aerospace researchers have placed the threshold of space at lower altitudes than the USAF and NASA, so many X-15 pilots could also be considered as astronauts. The "aeropause" region, where space-equivalent conditions are first encountered, starts at an altitude of 19 miles above the Earth. Many X-15 pilots traveled through, and far above, the aeropause. Out of all the X-15 missions, two flights (by the same pilot) also qualified for the international FAI definition of a spaceflight by passing the 62.1 mile (100 km) mark.

No. perryman is quite right about not being able to re-enter, but if you want to know specifically about whether they could have reached orbit the answer is still no. The x-15 did rely on those stubby wings for lift to some degree and as the air got thinner with altitude the plane's glide path would start to feather into a parabolic maximum that would never peak high enough to reach orbital altitudes.

No, It didn't carry nearly enough fuel to acheive that. by the time the plane reaches space, the engines have already flamed out.

X-15 pilot Bill Dana stated in his book that the x-15 could have reached an altitude of over 100 miles, but could not be recovered safely from that altitude. This is from an air drop from a B-52. Joe Walker could have easily exceeded 400,000 feet in his record breaking flight, but the mission profile was 360,000, (safety first) so he lowered the trajectory so as not to exceed that number. He described the 5g pull out as "one big long squeeze." The X-15 did not have the capacity to reach orbital velocity on its own, however. There were plans made to put the x-15 on a rocket and boost it to higher speeds and altitudes, and simulations were run for reentry from orbit. I understand that at least one pilot was able to repeatedly demonstrate successful reentry on the x-15 simulator. A follow on project involved a delta wing x-15 and the x-20 dyna soar. X-15 pilot Joe Engle hand flew a reentry in the space shuttle, drawing on his experience in the X-15. Unfortunately, the X-15 and X-20 programs were cancelled, so we may never know what might have been. Certainly a lot of work would have been necessary to put the x-15 into orbit and recover it safely, but who knows what they might have been able to do if they had been allowed to try.