Kokrak, at 22-year-old native of Ohio, is at 3 under for the tournament and leads Alex Prugh, David Merkow and Derek Fathauer. Prugh and Merkow finished their second round, while Fathauer was stranded on the course as darkness suspended play.
A four-hour fog delay on Monday pushed back the start of that round and the tournament has been trying to catch-up ever since.
Trevor Murphy shot a 2-under 68 during Tuesday's second round and is the only other player heading into Wednesday under par.
'I've played in four of the last five Amateurs, but I never played that well,' said Kokrak, who played the Lake Course. 'I never came in with the confidence I have now. Making the Open just opened up all the doors for me the rest of the summer.'
Sam Saunders, the 20-year-old grandson of Arnold Palmer, will miss the cut for match play, ending the second round with a two-round total of 11 over.
The field will be trimmed to the low 64 scores following the second round of stroke play. Match play is scheduled to begin Wednesday on the Lakeside Course and runs through the 36-hole final on Sunday.