Campbell failed to finish in the top three at Prince's -- one of four courses hosting qualifying for the year's third major -- after shooting rounds of 68 and 72 to finish 4-under 140.
Sandelin missed out by two shots at Rye.
Campbell, who won his lone major at Pinehurst No. 2 in 2005, can still qualify for the British Open. He would have to be the leading player not already eligible among the top five at the French Open and Scottish Open over the next two weeks.
Six players, including English amateurs Craig Hinton and Tom Lewis, qualified Tuesday. Six more will advance when qualifying is completed on Wednesday.