I think the answer is almost certainly yes, an algebraic space can fail to have a universal map to a scheme (a "schemification"). I don't have a proof, but I think I know the right place to look for one (besides David Rydh's immediate surroundings).

If we can find two maps of schemes which do not have a coequalizer in the category of schemes, but do have a coequalizer in the category of algebraic spaces, then the coequalizer algebraic space will not have a schemification.

Consider Hironaka's example of a non-projective proper variety (see page 15 of Knutson's Algebraic Spaces). It has an action of ℤ/2 for which there is an algebraic space quotient. But in the category of schemes, there is no geometric quotient for this action. The question is whether there is a categorical quotient in this case.

I think the answer is almost certainly yes, an algebraic space can fail to have a universal map to a scheme (a "schemification"). I don't have a proof, but I think I know the right place to look for one (besides David Rydh's immediate surroundings).

If we can find two maps of schemes which do not have a coequalizer in the category of schemes, but do have a coequalizer in the category of algebraic spaces, then the coequalizer algebraic space will not have a schemification.

Consider Hironaka's example of a non-projective proper variety (see page 15 of Knutson's Algebraic Spaces). It has an action of ℤ/2 for which there is an algebraic space quotient. But in the category of schemes, there is no geometric quotient for this action. The question is whether there is a categorical quotient in this case.