I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.
It's used in Google maps to introduce cache locality: When you move a little bit while viewing the map, you want to be moving only a little bit in the memory, which is after all arranged linearly. This is where continuous space filling curves find one of their uses.
Edit: You can find a non-academic references about Google's use of Hilbert curves.
In a very different style, N. Katz has studied "space filling curves" over finite fields (https://web.math.princeton.edu/~nmk/spacefill.pdf), and given some applications (e.g., every abelian variety over a finite field is a quotient of the jacobian of a curve, using ideas of Gabber).
Space filling curves have been used a good deal by Katsuya Eda and his coauthors to compute singular homology groups of Peano continua (which are precisely the continuous images of $[0,1]$).
The singular homology of the Griffiths Twin cone is computed in this paper. A key part of this is Theorem 3.1 which requires the use of a space filling curve.
Similar techniques are used in
K. Eda, K. Kawamura, The surjectivity of the canonical homomorphism from singular homology to Cech homology, Proc. Amer. Math. Soc. 128 (1999), 1487–1495. (See Lemma 2.4)
K. Eda, Singular homology groups of one-dimensional Peano continua (see Lemma 4.1)
In a similar manner to the way they are used by Google maps, space filling curves are used for load balancing super computers. Since computational domains are sometimes more refined in certain locations, space filling curves can catch this (by refining the curve as needed). Another advantage of using these curves for load balancing is that communication between computation units is usually better (though this is only a heuristic).