It seems that usually (by which I mean, in every source I've looked at) people define the group $K^0(X)$ for $X$ compact Hausdorff. Sometimes they later extend this definition to all locally compact Hausdorff spaces $X$ by defining $K^0(X)$ to be $K^0(X^+)$ of the one-point compactification $X^+$
The question is: why not just take the original definition of $K^0(X)$ for all locally compact Hausdorff $X$? Isomorphism classes of vector bundles on any such $X$ forms a commutative monoid, and so we should be able to take the Grothendieck group just the same, no?
The reason we don't use the same definition of $K^0$ for non-compact spaces as we do for compact spaces, is because the resulting set of functors would not form a cohomology theory.
Source: A comment of this MO post. Thanks to Dmitri Pavlov for pointing to this reference.